LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

GIKT  OK 


Accession         90255  Class 


Teachers'  Manual 
to  Walsh's  Primary 
Arithmetic* 


MATHEMATICS  FOR  COMMON  SCHOOLS 

A 

MANUAL   FOE   TEACHEES 


INCLUDING 


DEFINITIONS,  PEINCIPLES,  AND  EULES 

AND   SOLUTIONS   OF  THE   MOEE 

DIFFICULT   PEOBLEMS 


BY 


JOHN  H.  WALSH 

ASSOCIATE  SUPERINTENDENT  OF  PUBLIC  INSTRUCTION 
BROOKLYN,  N.Y. 


PRIMARY   ARITHMETIC 


D.  C.  HEATH  &  CO.,  PUBLISHEES . 
1900 


_  A3  / 

M/3 


COPYRIGHT,  189% 

BY  JOHN  H.  WALSH. 


J.  S.  Gushing  &  Co.  —  Berwick  &  Smith. 
Boston,  Mass.,  U.S.A. 


CONTENTS 

(PRIMARY   AND   ELEMENTARY   ARITHMETIC   MANUAL.) 


•>•  PAGB 

INTRODUCTORY 1 

Plan  and  scope  of  the  work  —  Grammar  school  algebra  —  Con- 
structive geometry. 

II 

GENERAL  HINTS 5 

Division  of  the  work  —  Additions  and  omissions  —  Oral  and 
written  work  —  Use  of  books  —  Conduct  of  the  recitation  —  Drills 
and  sight  work  —  Definitions,  principles,  and  rules — Language 
—  Analyses  —  Objective  illustrations  —  Approximate  answers  — 
Indicating  operations  —  Paper  vs.  slates. 

Ill 

EARLY  ARITHMETIC  TEACHING 13 

Counting  —  Primary  arithmetic  —  The  Grube  method  —  Slate 
problems. 

IV 

NOTES  ON  CHAPTER  ONE       .        .       .       .     .* 17 

V 

NOTES  ON  CHAPTER  Two  .       t 22 

VI 

NOTES  ON  CHAPTER  THREE   .        ....        •       •       .       .    27 

iii 


IV  CONTENTS 


VII 
NOTES  ON  CHAPTER  FOUB 32 

VIII 
NOTES  ON  CHAPTER  FIVE 41 

ANSWERS .  1 


MANUAL   FOR  TEACHEES 


INTRODUCTORY 

Plan  and  Scope  of  the  Work,  —  In  addition  to  the  subjects 
generally  included  in  the  ordinary  text-books  in  arithmetic, 
Mathematics  for  Common  /Schools  contains  such  simple  work 
in  algebraic  equations  and  constructive  geometry  as  can  be 
studied  to  advantage  by  pupils  of  the  elementary  schools. 

The  arithmetical  portion  is  divided  into  thirteen  chapters, 
each  of  which,  except  the  first,  contains  the  work  of  a  term  of 
five  months.  The  following  extracts  from  the  table  of  contents 
will  show  the  arrangement  of  topics : 

FIRST  AND  SECOND  YEARS 

Chapter  I,  —  Numbers  of  Three  Figures.  Addition  and  Sub- 
traction. 

THIRD  YEAR 

Chapters  II,  and  III,  —  Numbers  of  Five  Figures.  Multipli- 
ers and  Divisors  of  One  Figure.  Addition  and  Subtraction  of 
Halves,  of  Fourths,  of  Thirds.  Multiplication  by  Mixed  Num- 
bers. Pint,  Quart,  and  Gallon ;  Ounce  and  Pound.  Roman 
Notation. 

1 


2  MANUAL   FOR  TEACHERS 


FOURTH  YEAR 

Chapters  IV,  and  V,  —  Numbers  of  Six  Figures.  Multipliers  and 
Divisors  of  Two  or  More  Figures.  Addition  and  Subtraction  of 
Easy  Fractions.  Multiplication  by  Mixed  Numbers.  Simple 
Denominate  Numbers.  Roman  Notation. 


FIFTH  YEAR 

Chapters  VI,  and  VII.  —  Fractions.     Decimals  of  Three  Places. 
Bills.     Denominate  Numbers.     Simple  Measurements. 


SIXTH  YEAR 

Chapters  VIII,  and  IX,  —  Decimals.     Bills.     Denominate  Num- 
bers.    Surfaces  and  Volumes.     Percentage  and  Interest. 


SEVENTH  YEAR 

Chapters  XI,  and  XII,  —  Percentage  and  Interest.  Commercial 
and  Bank  Discount.  Cause  and  Effect.  Partnership.  Bonds 
and  Stocks.  Exchange.  Longitude  and  Time.  Surfaces  and 
Volumes. 

EIGHTH  YEAR 

Chapters  XIII,  and  XIV,  —  Partial  Payments.  Equation  of 
Payments.  Annual  Interest.  Metric  System.  Evolution  and 
Involution.  Surfaces  and  Volumes. 


INTRODUCTORY  3 

While  all  of  the  above  topics  are  generally  included  in  an 
eight  years'  course,  it  may  be  considered  advisable  to  omit  some 
of  them,  and  to  take  up,  instead,  during  the  seventh  and  eighth 
years,  the  constructive  geometry  work  of  Chapter  XVI.  Among 
the  topics  that  may  be  dropped  without  injury  to  the  pupil  are 
Bonds  and  Stocks,  Exchange,  Partial  Payments,  and  Equation 
of  Payments. 

Grammar  School  Algebra,  —  Chapter  X.,  consisting  of  a  dozen 
pages,  is  devoted  to  the  subject  of  easy  equations  of  one  unknown 
quantity,  as  a  preliminary  to  the  employment  of  the  equation  in 
so  much  of  the  subsequent  work  in  arithmetic  as  is  rendered 
more  simple  by  this  mode  of  treatment.  To  teachers  desirous 
of  dispensing  with  rules,  sample  solutions  of  type  examples,  etc., 
the  algebraic  method  of  solving  the  so-called  "  problems  "  in  per- 
centage, interest,  discount,  etc.,  is  strongly  recommended. 

In  Chapter  XV.,  intended  chiefly  for  schools  having  a  nine 
years'  course,  the  algebraic  work  is  extended  to  cover  simple 
equations  containing  two  or  more  unknown  quantities,  and  pure 
and  affected  quadratic  equations  of  one  unknown  quantity. 

No  attempt  has  been  made  in  these  two  chapters  to  treat 
algebra  as  a  science ;  the  aim  has  been  to  make  grammar-school 
pupils  acquainted,  to  some  slight  extent,  with  the  great  instru- 
ment of  mathematical  investigation,  —  the  equation. 

Constructive  Geometry,  —  Progressive  teachers  will  appreciate  the 
importance  of  supplementing  the  concrete  geometrical  instruction 
now  given  in  the  drawing  and  mensuration  work.  Chapter  XVI. 
contains  a  series  of  problems  in  construction  so  arranged  as  to 
enable  pupils  to  obtain  for  themselves  a  working  knowledge  of 
all  the  most  important  facts  of  geometry.  Applications  of  the 
facts  thus  ascertained,  are  made  to  the  mensuration  of  surfaces 
and  volumes,  the  calculation  of  heights  and  distances,  etc.  No 
attempt  is  made  to  anticipate  the  work  of  the  high-school  by 
teaching  geometry  as  a  science. 


4  MANUAL   FOR   TEACHERS 

While  the  construction  problems  are  brought  together  into  a 
single  chapter  at  the  end  of  the  book,  it  is  not  intended  that 
instruction  in  geometry  should  be  delayed  until  the  preceding 
work  is  completed.  Chapter  XVI.  should  be  commenced  not  later 
than  the  seventh  year,  and  should  be  continued  throughout  the 
remainder  of  the  grammar-school  course.  For  the  earlier  years, 
suitable  exercises  in  the  mensuration  of  the  surfaces  of  triangles 
and  quadrilaterals,  and  of  the  volumes  of  right  parallelopipedons 
have  been  incorporated  with  the  arithmetic  work. 


II 

GENERAL  HINTS 

Division  of  the  Work,  —  The  five  chapters  constituting  Part  I. 
of  Mathematics  for  Common  Schools  should  be  completed  by  the 
end  of  the  fourth  school  year.  The  remaining  eight  arithmetic 
chapters  constitute  half-yearly  divisions  for  the  second  four  years 
of  school.  Chapter  I.,  with  the  additional  oral  work  needed  in 
the  case  of  young  pupils,  will  occupy  about  two  years ;  the  re- 
maining four  chapters  should  not  take  more  than  half  a  year  each. 
When  the  Grube  system  is  used,  and  the  work  of  the  first  two 
years  is  exclusively  oral,  it  will  be  possible,  by  omitting  much  of 
the  easier  portions  of  the  first  two  chapters,  to  cover,  during  the 
third  year,  the  ground  contained  in  Chapters  I.,  II.,  and  III. 

Additions  and  Omissions,  —  The  teacher  should  freely  supple- 
ment the  work  of  the  text-book  when  she  finds  it  necessary  to  do 
so ;  and  she  should  not  hesitate  to  leave  a  topic  that  her  pupils 
fully  understand,  even  though  they  may  not  have  worked  all  the 
examples  given  in  connection  therewith.  A  very  large  number 
of  exercises  is  necessary  for  such  pupils  as  can  devote  a  half-year 
to  the  study  of  the  matter  furnished  in  each  chapter.  In  the 
case  of  pupils  of  greater  maturity,  it  will  be  possible  to  make 
more  rapid  progress  by  passing  to  the  next  topic  as  soon  as  the 

previous  work  is  fairly  well  understood. 

• 

Oral  and  Written  Work,  — The  heading  "Slate  Problems"  is 
merely  a  general  direction,  and  it  should  be  disregarded  by  the 
teacher  when  the  pupils  are  able  to  do  the  work  "  mentally." 
The  use  of  the  pencil  should  be  demanded  only  so  far  as  it  may 

5 


6  MANUAL   FOR  TEACHERS 


be  required.  It  is  a  pedagogical  mistake  to  insist  that  all  of  the 
pupils  of  a  class  should  set  down  a  number  of  figures  that  are 
not  needed  by  the  brighter  ones.  As  an  occasional  exercise,  it 
may  be  advisable  to  have  scholars  give  all  the  work  required  to 
solve  a  problem,  and  to  make  a  written  explanation  of  each  step 
in  the  solution ;  but  it  should  be  the  teacher's  aim  to  have  the 
majority  of  the  examples  done  with  as  great  rapidity  as  is  con- 
sistent with  absolute  correctness.  It  will  be  found  that,  as  a 
rule,  the  quickest  workers  are  the  most  accurate. 

Many  of  the  slate  problems  can  be  treated  by  some  classes  as 
"  sight "  examples,  each  pupil  reading  the  question  for  himself 
from  the  book,  and  writing  the  answer  at  a  given  signal  without 
putting  down  any  of  the  work. 

Use  of  Books.  —  It  is  generally  recommended  that  books  be 
placed  in  pupils'  hands  as  early  as  the  third  school  year.  Since 
many  children  are  unable  at  this  stage  to  read  with  sufficient 
intelligence  to  understand  the  terms  of  a  problem,  this  work 
should  be  done  under  the  teacher's  direction,  the  latter  reading 
the  questions  while  the  pupils  follow  from  their  books.  In  later 
years,  the  problems  should  be  solved  by  the  pupils  from  the 
books  with  practically  no  assistance  whatever  from  the  teacher. 

Conduct  of  the  Eecitation,  —  Many  thoughtful  educators  consider 
it  advisable  to  divide  an  arithmetic  class  into  two  sections,  for 
some  purposes,  even  where  its  members  are  nearly  equal  in 
attainments.  The  members  of  one  division  of  such  a  class  may 
work  examples  from  their  books  while  the  others  write  the 
answers  to  oral  problems  given  by  the  teacher,  etc. 

Where  a  class  is  thus  taught  in  two  divisions,  the  members  of 
each  should  sit  in  alternate  rows,  extending  from  the  front 
of  the  room  to  the  rear.  Seated  in  this  way,  a  pupil  is  doing  a 
different  kind  of  work  from  those  on  the  right  and  the  left,  and 
he  would  not  have  the  temptation  of  a  neighbor's  slate  to  lead 
him  to  compare  answers. 


GENERAL   H 

As  an  economy  of  time,  explanations  of  new  subjects  might  be 
given  to  the  whole  class;  but  much  of  the  arithmetic  work 
should  be  done  in  "sections,"  one  of  which  is  under  the  im- 
mediate direction  of  the  teacher,  the  other  being  employed 
in  "seat"  work.  In  the  case  of  pupils  of  the  more  advanced 
classes,  "seat"  work  should  consist  largely  of  "problems"  solved 
without  assistance.  Especial  pains  have  been  taken  to  so  grade 
the  problems  as  to  have  none  beyond  the  capacity  of  the  average 
pupil  that  is  willing  to  try  to  understand  its  terms.  It  is  not 
necessary  that  all  the  members  of  a  division  should  work  the 
same  problems  at  a  given  time,  nor  the  same  number  of  prob- 
lems, nor  that  a  new  topic  should  be  postponed  until  all  of  the 
previous  problems  have  been  solved. 

Whenever  it  is  possible,  all  of  the  members  of  the  division 
working  under  the  teacher's  immediate  direction  should  take 
part  in  all  the  work  done.  In  mental  arithmetic,  for  instance, 
while  only  a  few  may  be  called  upon  for  explanations,  all  of  the 
pupils  should  write  the  answers  to  each  question.  The  same  is 
true  of  much  of  the  sight  work,  the  approximations,  some  of  the 
special  drills,  etc. 

Drills  and  Sight  Work.  —  To  secure  reasonable  rapidity,  it  is 
necessary  to  have  regular  systematic  drills.  They  should  be 
employed  daily,  if  possible,  in  the  earlier  years,  but  should  never 
last  longer  than  five  or  ten  minutes.  Various  kinds  are  sug- 
gested, such  as  sight  addition  drills,  in  Arts.  3,  11,  24,  26,  etc. ; 
subtraction,  in  Arts.  19,  50,  53,  etc. ;  multiplication,  in  Arts.  71, 
109,  etc. ;  division,  in  Arts.  199,  202,  etc. ;  counting  by  2's,  3's, 
etc.,  in  Art.  61 ;  carrying,  in  Art.  53,  etc.  For  the  young  pupil, 
those  are  the  most  valuable  in  which  the  figures  are  in  his  sight, 
and  in  the  position  they  occupy  in  an  example ;  see  Arts.  3,  34, 
164,  etc. 

Many  teachers  prepare  cards,  each  of  which  contains  one  of 
the  combinations  taught  in  their  respective  grades.  Showing 
one  of  these  cards,  the  teacher  requires  an  immediate  answer 


8  MANUAL  FOR  TEACHERS 


from  a  pupil.  If  his  reply  is  correct,  a  new  card  is  shown  to 
the  next  pupil,  and  so  on.  Other  teachers  write  a  number  of 
combinations  on  the  blackboard,  and  point  to  them  at  random, 
requiring  prompt  answers.  When  drills  remain  on  the  board 
for  any  considerable  time,  some  children  learn  to  know  the 
results  of  a  combination  by  its  location  on  the  board,  so  that 
frequent  changes  in  the  arrangement  of  the  drills  are,  therefore, 
advisable.  The  drills  in  Arts.  Ill,  112,  and  115  furnish  a  great 
deal  of  work  with  the  occasional  change  of  a  single  figure. 

For  the  higher  classes,  each  chapter  contains  appropriate 
drills,  which  are  subsequently  used  in  oral  problems.  It  happens 
only  too  frequently  that  as  children  go  forward  in  school  they 
lose  much  of  the  readiness  ia  oral  and  written  work  they 
possessed  in  the  lower  grades,  owing  to  the  neglect  of  their 
teachers  to  continue  to  require  quick,  accurate  review  work  in 
the  operations  previously  taught.  These  special  drills  follow 
the  plan  of  the  combinations  of  the  earlier  chapters,  but  gradu- 
ally grow  more  difficult.  They  should  first  be  used  as  sight 
exercises,  either  from  the  books  or  from  the  blackboard. 

To  secure  valuable  results  from  drill  exercises,  the  utmost 
possible  promptness  in  answers  should  be  insisted  upon. 

Definitions,  Principles,  and  Eulee,  — Young  children  should  not 
memorize  rules  or  definitions.  They  should  learn  to  add  by 
adding,  after  being  first  shown  by  the  teacher  how  to  perform 
the  operation.'  Those  not  previously  taught  by  the  Grube 
method  should  be  given  no  reason  for  "  carrying."  In  teaching 
such  children  to  write  numbers  of  two  or  three  figures,  there  is 
nothing  gained  by  discussing  the  local  value  of  the  digits.  Dur- 
ing the  earlier  years,  instruction  in  the  art  of  arithmetic  should 
be  given  with  the  least  possible  amount  of  science.  While  prin- 
ciples may  be  incidentally  brought  to  the  view  of  the  children 
at  times,  there  should  be  no  cross-examination  thereon.  It  may 
be  shown,  for  instance,  that  subtraction  is  the  reverse  of  addition, 
and  that  multiplication  is  a  short  method  of  combining  equal 


GENERAL   HINTS  9 

numbers,  etc. ;  but  care  should  be  taken  in  the  case  of  pupils 
below  about  the  fifth  school  year  not  to  dwell  long  on  this  side 
of  the  instruction.  By  that  time,  pupils  should  be  able  to  add, 
subtract,  multiply,  and  divide  whole  numbers ;  to  add  and  sab- 
tract  simple  mixed  numbers,  and  to  use  a  mixed  number  as  a 
multiplier  or  a  multiplicand  ;  to  solve  easy  problems,  with  small 
numbers,  involving  the  foregoing  operations  and  others  contain- 
ing the  more  commonly  used  denominate  units.  Whether  or  not 
they  can  explain  the  principles  underlying  the  operations  is  of 
next  to  no  importance,  if  they  can  do  the  work  with  reasonable 
accuracy  and  rapidity. 

When  decimal  fractions  are  taken  up,  the  principles  of  Arabic 
notation  should  be  developed  ;  and  about  the  same  time,  or  some- 
what later,  the  principles  upon  which  are  founded  the  operations 
in  the  fundamental  processes,  can  be  briefly  discussed. 

Definitions  should  in  all  cases  be  made  by  the  pupils,  their 
mistakes  being  brought  but  by  the  teacher  through  appropriate 
questions,  criticisms,  etc.  Systematic  work  under  this  head 
should  be  deferred  until  at  least  the  seventh  year. 

The  use  of  unnecessary  rules  in  the  higher  grades  is  to  be 
deprecated.  When,  for  instance,  a  pupil  understands  that  per 
cent  means  hundredths,  that  seven  per  cent  means  seven  hun- 
dredths,  it  should  not  be  necessary  to  tell  him  that  7  per  cent  of 
143  is  obtained  by  multiplying  143  by  .07.  It  should  be  a  fair 
assumption  that  his  previous  work  in  the  multiplication  of 
common  and  of'  decimal  fractions  has  enabled  him  to  see  that 
7  per  cent  of  143  is  yfo  of  143  or  143  X  .07,  without  information 
other  than  the  meaning  of  the  term  "  per  cent." 

When  a  pupil  is  able  to  calculate  that  15  %  of  120  is  18,  he 
should  be  allowed  to  try  to  work  out  for  himself,  without  a  rule, 
the  solution  of  this  problem :  18  is  what  per  cent  of  120  ?  or  of 
this:  18  is  15%  of  what  number?  These  questions  should 
present  no  more  difficulty  in  the  seventh  year  than  the  following 
examples  in  the  fifth  :  (a)  Find  the  cost  of  ^  ton  of  hay  at  $12 
per  ton.  (b)  When  hay  is  worth  $12  per  ton,  what  part  of  a 


10  MANUAL   FOR   TEACHERS 

ton  can  be  bought  for  $  1.80  ?  (c)  If  ^  ton  of  hay  costs  $1.80, 
what  is  the  value  of  a  ton  ? 

When,  however,  it  becomes  necessary  to  assist  pupils  in  the 
solution  of  problems  of  this  class,  it  is  more  profitable  to  furnish 
them  with  a  general  method  by  the  use  of  the  equation,  than 
with  any  special  plan  suited  only  to  the  type  under  immediate 
discussion. 

In  the  supplement  to  the  Manual  will  be  found  the  usual  defini- 
tions, principles,  and  rules,  for  the  teacher  to  use  in  such  a  way 
as  her  experience  shows  to  be  best  for  her  pupils.  The  rules 
given  are  based  somewhat  on  the  older  methods,  rather  than  on 
those  recommended  by  the  author.  He  would  prefer  to  omit 
entirely  those  relating  to  percentage,  interest,  and  the  like  as 
being  unnecessary,  but  that  they  are  called  for  by  many  success- 
ful teachers,  who  prefer  to  continue  the  use  of  methods  which 
they  have  found  to  produce  satisfactory  results. 

Language.  —  While  the  use  of  correct  language  should  be 
insisted  upon  in  all  lessons,  children  should  not  be  required  in 
arithmetic  to  give  all  answers  in  "  complete  sentences."  Espe- 
cially in  the  drills,  it  is  important  that  the  results  be  expressed 
in  the  fewest  possible  words. 

Analyses,  —  Sparing  use  of  analyses  is  recommended  for  begin- 
ners. If  a  pupil  solves  a  problem  correctly,  the  natural  inference 
should  be  that  his  method  is  correct,  even  if  he  be  unable  to  state 
it  in  words.  When  a  pupil  gives  the  analysis  of  a  problem,  he 
should  be  permitted  to  express  himself  in  his  own  way.  Set 
forms  should  not  be  used  under  any  circumstances. 

Objective  Illustrations, —  The  chief  reason  for  the  use  of  objects 
in  the  study  of  arithmetic  is  to  enable  pupils  to  work  without 
them.  While  counters,  weights  and  measures,  diagrams,  or  the 
like  are  necessary  at  the  beginning  of  some  topics,  it  is  important 
to  discontinue  their  use  as  soon  as  the  scholar  is  able  to  proceed 
without  their  aid. 


GENERAL    HINTS  11 

Approximate  Answers,  —  An  important  drill  is  furnished  in 
the  "approximations."  (See  Arts.  521,  669,  719,  etc.)  Pupils 
should  be  required  in  much  of  their  written  work  to  estimate 
the  result  before  beginning  to  solve  a  problem  with  the  pencil. 
Besides  preventing  an  absurd  answer,  this  practice  will  also  have 
the  effect  of  causing  a  pupil  to  see  what  processes  are  necessary. 
In  too  many  instances,  work  is  commenced  upon  a  problem  before 
the  conditions  are  grasped  by  the  youthful  scholar ;  which  will 
be  less  likely  to  occur  in  the  case  of  one  who  has  carefully 
"  estimated  "  the  answer.  The  pupil  will  frequently  find,  also, 
that  he  can  obtain  the  correct  result  without  using  his  pencil 
at  all. 

Indicating  Operations,  —  It  is  a  good  practice  to  require  pupils 
to  indicate  by  signs  all  of  the  processes  necessary  to  the  solution 
of  a  problem,  before  performing  any  of  the  operations.  This  fre- 
quently enables  a  scholar  to  shorten  his  work  by  cancellation,  etc. 
In  the  case  of  problems  whose  solution  requires  tedious  processes, 
some  teachers  do  not  require  their  pupils  to  do  more  than  to 
indicate  the  operations.  It  is  to  be  feared  that  much  of  the  lack 
of  facility  in  adding,  multiplying,  etc.,  found  in  the  pupils  of 
the  higher  classes  is  due  to  this  desire  to  make  work  pleasant. 
Instead  of  becoming  more  expert  in  the  fundamental  operations, 
scholars  in  their  eighth  year  frequently  add,  subtract,  multiply, 
and  divide  more  slowly  and  less  accurately  than  in  their  fourth 
year  of  school. 

Paper  vs.  Slates,  —  To  the  use  of  slates  may  be  traced  very  much 
of  the  poor  work  now  done  in  arithmetic.  A  child  that  finds  the 
sum  of  two  or  more  numbers  by  drawing  on  his  slate  the  number 
of  strokes  represented  by  each,  and  then  counting  the  total,  will 
have  to  adopt  some  other  method  if  his  work  is  done  on  material 
that  does  not  permit  the  easy  obliteration  of  the  tell-tale  marks. 
When  the  teacher  has  an  opportunity  to  see  the  number  of 
attempts  made  by  some  of  her  pupils  to  obtain  the  correct  quo- 


12  MANUAL   FOR  TEACHERS 

tient  figures  in  a  long  division  example,  she  may  realize  the 
importance  of  such  drills  as  will  enable  them  to  arrive  more 
readily  at  the  correct  result. 

The  unnecessary  work  now  done  by  many  pupils  will  be  very 
much  lessened  if  they  find  themselves  compelled  to  dispense  with 
the  "rubbing  out"  they  have  an  opportunity  to  indulge  in  when 
slates  are  employed.  The  additional  expense  caused  by  the 
introduction  of  paper  will  almost  inevitably  lead  to  better  results 
in  arithmetic.  The  arrangement  of  the  work  will  be  looked 
after ;  pupils  will  not  be  required,  nor  will  they  be  permitted,  to 
waste  material  in  writing  out  the  operations  that  can  be  per- 
formed mentally ;  the  least  common  denominator  will  be  deter- 
mined by  inspection  ;  problems  will  be  shortened  by  the  greater 
use  of  cancellation,  etc.,  etc.  Better  writing  of  figures  and  neater 
arrangement  of  problems  will  be  likely  to  accompany  the  use  of 
material  that  will  be  kept  by  the  teacher  for  the  inspection  of 
the  school  authorities.  The  endless  writing  of  tables  and  the 
long,  tedious  examples  now  given  to  keep  troublesome  pupils 
from  bothering  a  teacher  that  wishes  to  write  up  her  records, 
will,  to  some  extent,  be  discontinued  when  slates  are  nr  longer 
-jsed. 


w  - 


III 

EARLY  ARITHMETIC  TEACHING 

Counting,  —  While  the  majority  of  children  are  able,  upon  enter- 
ing school,  to  repeat  the  names  of  the  first  ten  or  more  numbers, 
they  are  not  always  able  to  count  things.  The  first  duty  of  the 
teacher  is  to  secure  correct  notions  of  the  first  nine  numbers,  and 
this  can  best  be  done  by  the  employment  of  objects,  such  as  beans, 
splints,  shoe-pegs,  blocks,  etc.  A  numeral  frame  is  very  useful 
for  this  purpose. 

In  counting,  it  is  very  important  to  have  the  child  understand 
that  the  second  splint  is  not  two  splints.  This  may  be  made  clear 
to  a  child  by  having  him  put  on  his  desk  one  bean,  then  near  it 
two  beans,  three  beans  in  another  place,  etc.  After  the  pupil 
can  count  understandingly  to  nine,  he  should  be  taught  the 
figures.  The  notation  and  numeration  of  numbers  of  two  or 
more  figures  will  be  discussed  in  later  chapters. 

Primary  Arithmetic,  —  After  children  have  learned  to  count 
readily,  experts  disagree  as  to  the  best  method  of  procedure. 
Many  excellent  teachers  believe  that  work  should  be  commenced 
at  once  upon  numeration  and  notation,  followed  by  the  funda- 
mental operations  in  the  usual  order.  Some  of  the  advocates  of 
this  method  favor  the  completion  of  each  topic  before  proceeding 
to  the  next ;  that  is,  numeration  and  notation  are  taught  at  least 
to  billions ;  then  addition  is  taken  up,  beginning  with  small  num- 
bers and  gradually  increasing  to  examples  containing  numbers  of 
eight  or  nine  figures.  Subtraction,  multiplication,  and  division 
are  each  studied  to  this  extent  before  the  next  is  commenced. 

The  more  intelligent  advocates  of  teaching  operations  at  the 

IS 


14  MANUAL   FOR   TEACHERS 

outset,  recognize  the  fact  that  it  is  neither  necessary  nor  advisable 
to  defer  the  addition  of  small  numbers  until  children  are  able  to 
write  those  of  three  or  more  periods,  nor  to  postpone  finding  the 
sum  of  ^  and  ^  until  after  the  properties  of  numbers  have  been 
studied  in  the  fifth  school  year.  Their  plan  is  to  follow  such 
simple  examples  in  the  addition  of  small  numbers  as  involve  no 
carrying,  by  corresponding  ones  in  subtraction.  More  difficult 
examples  in  both  of  these  operations  come  next,  followed  by  sim- 
ple ones  in  multiplication  and  division.  Easy  work  in  fractions 
is  introduced  at  an  early  stage,  and  problems  involving  the  more 
common  denominate  units  are  brought  in  from  time  to  time. 

The  Grnbe  Method,  —  A  growing  number  of  educators  believe 
that  early  arithmetical  instruction  should  be  based  upon  the 
study  of  numbers,  rather  than  upon  that  of  processes,  —  that 
the  former  should  be  the  prominent  feature  of  the  early  instruc- 
tion, and  the  latter  incidental,  at  least  for  the  first  two  years. 

This  method,  called  after  its  inventor,  Grube,  requires  the 
teaching  of  all  of  the  processes  in  the  case  of  each  number  before 
proceeding  to  the  next.  Thus,  when  the  number  4  is  studied, 
the  pupil  measures  it  by  all  numbers  smaller  than  itself.  Using 
4  beans,  he  measures  by  1,  by  arranging  them  as  follows : 
0000.  In  this  way  he  sees  that  1  +  1  +  1  +  1  =  4;  that 
there  are  4  ones  in  4,  or  1  X  4  =  4;  that  4  —  1  —  1  —  1  =  1;  that 
4^1  =  4. 

Measuring  by  2,  00  00,  he  sees  that  2  +  2  =  4,  2x2  =  4, 
4-2=2,  4-^2  =  2. 

Measuring  by  3,  000  0,  he  sees  that  3+1  =  4,  1  +  3  =  4; 
4-3=1,  4-1  =  3;  that  (1  X  3)  +  1  =  4,  and  that  4  -*-  3  =  1 
and  1  over. 

The  pupil  then  answers  questions  given  by  the  teacher,  first 
using  the  counters  and  afterwards  without  them  :  — 

Four  is  how  many  more  than  3?  Than  1?  Than  2?  Three 
is  how  many  less  than  4  ?  Two  is  how  many  less  ?  One  is  how 
many  less  ? 


EARLY  ARITHMETIC   TEACHING  15 

How  many  ones  in  4  ?  How  many  twos  ?  Threes  ?  One-half 
of  4  is  what ?  Two  is  %  of  what  number? 

Problems  containing  the  foregoing  combinations  are  then  given 
in  great  variety  by  the  teacher  until  all  of  the  facts  about  the 
number  4  in  its  relation  with  the  smaller  numbers  are  fully 
mastered. 

In  teaching  any  number,  no  larger  number  must  appear  in 
any  way  whatever.  During  the  study  of  4,  it  is  not  permissible 
to  ask  4  twos,  or  that  4  is  1  less  than  what,  etc.,  etc. 

The  work  proceeds  slowly  and  thoroughly,  at  least  a  year 
being  devoted  to  the  numbers  below  10.  The  second  year  is 
given  to  the  numbers  from  10  to  20,  and  the  third  year  to  those 
from  20  to  100.  This  is  probably  as  far  as  the  method  is  carried 
in  this  country. 

In  the  greater  number  of  the  schools  using  this  method, 
systematic  instruction  in  the  fundamental  processes  is  commenced 
by  the  beginning  of  the  third  year ;  while  in  some,  the  Grube 
method  is  used  for  oral  work,  and  the  teaching  of  slate  addition 
is  carried  on  at  the  same  time,  even  during  the  first  year. 

Slate  Problems,  —  When,  instead  of  receiving  oral  instruction 
for  some  time,  children  are  taught  processes  from  the  outset,  it 
frequently  happens  that  many  of  them  show  little  ability  in 
solving  problems.  While  some  attention  should  be  given  in  the 
early  years  to  this  side  of  arithmetic,  it  should  not  be  permitted 
to  retard  too  much  the  advancement  of  pupils.  Many  of  them 
have  to  leave  school  soon,  and  they  should  be  taught  as  rapidly 
as  is  consistent  with  real  progress  to  perform  accurately  the 
ordinary  operations  in  whole  numbers,  simple  fractions,  and 
decimals.  Being  familiar  with  these  tools,  greater  maturity 
will,  of  itself,  show  which  is  to  be  used  in  such  questions  as  are 
likely  to  come  up  in  ordinary  avocations. 

The  teacher  should  exercise  much  care  to  give  only  such 
problems  as  can  readily  be  understood  by  the  pupil,  and  which 
do  not  contain  too  many  conditions  or  numbers  that  bewilder 


16  MANUAL   FOR  TEACHERS 

the  learner.  While  a  beginner  will  have  no  difficulty  in  deter- 
mining whether  to  add  or  subtract  in  a  mental  problem  suited 
to  his  capacity,  the  same  kind  of  problem  with  larger  figures 
will  give  him  much  difficulty.  For  this  reason,  the  earlier  slate 
problems  should  be  the  merest  trifle  beyond  his  ability  to  solve 
mentally.  In  his  attempt  to  work  them  out  in  his  head,  he  will 
determine  whether  addition  or  subtraction  is  needed,  etc. 

Problems  in  all  grades  should  be  "miscellaneous,"  and  pupils 
should  be  allowed  as  far  as  possible  to  determine  for  themselves 
what  operation  is  necessary  to  solve  any  given  one. 


IV 

NOTES  ON  CHAPTER  ONE 

THE  hints  given  as  to  the  work  of  this  chapter  are  intended 
chiefly  for  the  guidance  of  teachers  of  young  children  that  are 
beginning  slate  work  in  the  fundamental  processes  without  much 
preliminary  oral  instruction.  Pupils  that  have  been  taught  for 
two  years  by  the  Grube  method  should  not  be  required  to  spend 
unnecessary  time  on  the  simpler  portions  of  the  work. 

Art.  4.  —  In  teaching  notation  of  numbers  of  two  figures  to 
young  children  that  have  not  been  previously  taught  by  the 
Grube  method,  it  is  not  advisable  to  lay  stress  on  the  local  value 
of  the  tens'  figure.  Show  them  how  to  read  and  write  10,  11, 
12,  etc.,  to  20;  then  30,  40,  50,  etc.,  to  90.  After  this,  there  is 
but  little  difficulty. 

7.  By  working  an  example  for  the  pupils,  teach  them  to  place 
under  each  column  its  sum.  As  their  tendency  is  to  begin  work- 
ing at  the  left,  be  careful  to  see  that  they  always  commence  to 
add  at  the  right. 

9.  The  problems  will  present  no  difficulty,  as  they  involve 
only  addition. 

11.  These  sight  exercises  may  first  be  employed  as  drills  to 
teach  children  to  use  in  blackboard  addition  as  few  words  as 
possible.  The  first  figure  should  not  be  named,  —  only  the  sum 
of  the  first  and  the  second,  then  this  total  added  to  the  third. 
In  subsequent  drills  upon  these  combinations,  each  pupil  should, 
in  turn,  give  the  sum  of  any  set  indicated  by  the  teacher.  The 
work  should  be  done  rapidly  to  be  of  value. 

17 


18  MANUAL    FOR   TEACHERS 

13.  The  making  of  original  problems  by  the  pupils  should  be 
a  feature  of  every  grade. 

15  and  16.  Subtraction  is  here  introduced  by  the  "  building- 
up  "  method.  Pupils  find  it  easier  to  ascertain  the.  difference 
between  two  numbers  by  going  forward  from  the  smaller  to  the 
larger,  than  by  "  taking  away  "  one  from  the  other. 

17  consists  of  sight  exercises  in  the  form  of  addition,  leading 
to  the  subtraction  exercises  in  Art.  19. 

21.  While  in  adding,  the  use  of  the  word  and  is  considered 
unnecessaiy  ;  in  subtracting,  it  is  used  just  before  the  figure  that 
is  to  be  written. 

For  some  advantages  obtained  by  employing  the  "  building- 
up  "  method,  see  Art.  384,  where  it  is  used  to  obtain  in  one  oper- 
ation the  difference  between  1000  and  643  +  287  -f  25.  In  Art. 
385,  it  is  used  to  find  a  remainder  in  long  division  without  writ- 
ing the  product  of  the  divisor  by  the  quotient. 

23.  Here  begins  the  real  problem  work,  as  the  pupil  has  now 
to  determine  for  the  first  time  in  slate  examples  whether  the 
result  is  to  be  reached  by  addition  or  subtraction.  When  the 
pupils  are  able  to  solve  one  of  these  problems  without  using 
the  pencil,  it  should  be  repeated,  but  with  such  a  change  in  one 
of  the  numbers  as  will  render  necessary  the  use  of  the  slate. 
For  the  10  cents  in  the  first  example,  for  instance,  14  cents  or  24 
cents  may  be  substituted. 

As  many  pupils  attend  rather  to  the  numbers  in  a  problem 
than  to  its  terms,  some  may  subtract  when  they  should  add, 
especially  as  this  seems  the  natural  operation  when  only  two 
numbers  are  involved.  It  is  important  that  they  should  be  led 
to  see  that  the  size  of  the  numbers  does  not  change  the  nature  of 
the  example,  and  that  they  can  easily  determine  whether  addi- 
tion or  subtraction  is  required,  by  considering  what  operation 


NOTES   ON  CHAPTER  ONE  19 

they  would  employ  in  a  similar  example  containing  very  small 
figures. 

It  is  not  advisable  as  a  regular  thing  to  follow  an  oral  problem 
by  a  written  one  of  exactly  the  same  nature,  as  this  tends  to 
make  children  inattentive  to  the  terms  of  the  latter  when  they 
already  know  from  the  oral  problem  what  operation  is  required. 

28.  It  is  inadvisable  to  waste  time  in  endeavoring  to  make 
clear  to  very  young  children  the  reason  for  "  carrying." 

37.  Teachers  should  require  pupils  to  write  the  proper  sign 
before  working  an  example,  as  this  tends  to  make  them  listen 
more  carefully  in  order  to  determine  whether  addition  or  sub- 
traction is  involved.     In  some  problems  that  are  too  simple  to 
need  the  use  of  the  pencil,  changes  may  be  made  in  the  numbers 
employed ;  great  care,  however,  should  be  taken  not  to  use  num- 
bers so  large  as  to  confuse  the  pupils. 

38.  Have  children  uriderstand  that  when  a  number  contains 
the  word  "  hundred,"  it  should  consist  of  three  figures.     Do  not 
explain. 

54.  These  exercises  are  intended  to  lead  up  to  the  subtraction 
with  "  borrowing  "  in  the  next  article.     Perhaps  the  following 
would  be  a  better  arrangement : 

?  ?  ?  ?  ?  ? 

+  29         +37         +17         +86         +   75         +   90 
41  50  25  90  100  150 

As  children  are  generally  taught  to  begin  with  the  bottom 
figure  in  addition,  they  will  naturally  say  in  the  first  example, 
9  and  2  are  11,  writing  the  2  in  its  place,  etc. 

55.  Subtraction  with  "  borrowing  "  is  generally  taught  in  one 
of  three  ways.     The  "building-up"  method   given  in  the  text 
is  the  most  readily  taken  hold  of  by  young  pupils. 


20  MANUAL   FOR  TEACHERS 

**          By  the   second  method,  the  child  is  instructed  that 
whenever  he  increases  by  ten  any  figure  of  the  minuend, 

g9       he  must  add  1  to  the  next  figure  of  the  subtrahend. 
Seeing  that  he  cannot  take  9  from  1,  he  says  9  from  11 
leaves  2 ;  1  (to  carry)  and  2  are  3,  3  from  4  leaves  1. 

While  this  method  is  just  as  logical  as  the  next,  it  is  not  so 
easily  "  explained,"  and,  for  this  reason,  is  not  so  much  favored 
by  many  teachers  of  the  present  day. 

The  third  method  consists  of  diminishing  the  next  left-      sii 
hand  figure  of  the  minuend  after  "  borrowing."     Where 
the  minuend  contains  ciphers,  this  method  is  particularly      — 
confusing  to  beginners,  especially  where  they  are  forbidden,  as 
should  be  the  case,  to  write  the  changes  that  are  made  in  the 
figures  of  the  minuend. 

Except  in  the  addition  of  long  columns,  children  should  be 
required  from  the  beginning  of  slate  work  to  abstain  from  count- 
ing, writing  "  carrying  "  figures,  and  the  like.  The  guide  fig- 
ures introduced  into  the  foregoing  explanations  of  methods  of 
subtracting  should  not  be  used  by  pupils. 

61.  As  a  change  from  sight  work,  and  to  increase  the  pupils' 
readiness  in  the  solution  of  mental  examples,  these  drills  are 
useful.  Not  requiring  any  preliminary  writing  on  the  board, 
they  can  be  taken  up  at  any  time  the  class  is  unoccupied  for  a 
few  minutes  —  waiting  for  the  signal  to  go  home,  for  example. 

The  pupils  all  stand ;  the  teacher  announces  the  number  to 
be  added,  2  for  instance,  and  begins  by  saying  1  herself.  The 
first  pupil  says  3,  then  sits ;  the  next,  5  ;  and  so  on.  After  39, 
or  some  other  convenient  number,  is  reached,  the  teacher  begins 
by  saying  2,  and  the  pupils,  in  order,  give  4,  6,  8,  etc.,  to  40. 

The  intelligent  teacher  will  be  careful  to  suit  these  drills  to  the 
capacity  of  her  pupils.  She  will  not  weary  beginners  by  spend- 
ing too  much  time  on  the  more  difficult  drills  with  7,  8,  and  9 ; 
nor  will  she  waste  the  time  of  older  scholars  by  dwelling  on  the 
addition  by  twos. 


NOTES   ON   CHAPTER  ONE  21 

The  same  kind  of  work  may  be  employed  as  subtraction  drills. 

Subtract  by  twos : 

40,  38,  36,  etc. 

39,  37,  35,  etc. 

By  threes : 

40,  37,  34,  etc. 

39,  36,  33,  etc. 

38,  35,  32,  etc. 
By  fours : 

40,  36,  32,  etc. 

39,  35,  31,  etc. 
38,  34,  30,  etc. 
37,  33,  29,  etc. 


V 

NOTES  ON  CHAPTER  TWO 

74.  Slate  multiplication  is  commenced  as  soon  as  the  table  of 
2  times  is  learned.  The  first  examples  contain  no  carrying. 

76.   Division  tables  should  not  be  memorized. 

81.  Do  not  permit  children  to  prefix  an  unnecessary  cipher 
in  the  quotient  of  100^-2;  that  is,  do  not  have  the  answer 
written  050. 

84.  Many  scholars  think  that  when  a  slate  problem  contains  a 
very  small  number  and  a  large  one,  they  must  either  multiply 
or  divide.  Examples  1—4  are  given  with  simple  numbers  to 
show  them  that  the  nature  of  the  operation  depends  entirely 
upon  the  conditions  of  a  problem.  While  pupils  should  not  be 
required  to  use  a  pencil  to  solve  a  problem  that  can  be  solved 
mentally,  it  would  help  the  class  to  have  these  four  examples 
worked  on  the  board  as  an  indication  that  in  the  subsequent 
examples  there  may  be  needed  any  one  of  the  four  operations 
learned  thus  far,  and  to  serve  as  a  model  in  their  arrangement 
of  the  other  problems. 

While  many  teachers  require  the  pupils  to  write  the  denomina- 
tion of  each  addend,  of  the  subtrahend  and  the  minuend,  of  the 
multiplicand,  and  of  the  dividend,  it  is  scarcely  necessary.  In 
later  life  it  is  not  done ;  and  confusion  is  sometimes  produced  in 
the  minds  of  young  scholars  by  attempting  to  make  them  under- 
stand why,  for  example,  60  pints  divided  by  2  will  sometimes 
give  a  quotient  of  30  pints,  and  at  other  times,  as  in  the  6th 


UHIT'EKSITT 

NOTES   ON   CHAPTES^gSfKfcALlFQSS^          23 


problem,  an  apparent  quotient  of  30  quarts.  It  will  be  found 
more  satisfactory,  even  if  less  scientific,  to  have  the  denomination 
written  only  with  the  result. 

Although  no  formal  instruction  in  finding  halves  and  thirds  of 
numbers  has  as  yet  been  given,  the  average  pupil  will  be  able  to 
solve  problems  10,  11,  and  14. 

85.  Lay  no  stress  on  the  local  value  of  the  figures.  Practice 
will  enable  the  children  to  read  and  write  correctly  numbers 
of  four  figures.  Teach  the  pupil  to  write  the  comma  when  the 
word  "  thousand  "  is  said  and  after  the  number  of  thousands,  the 
comma  to  be  followed  always  by  three  figures. 

97.  Children  should  be  led  to  see  that  12  X  2  is  the  same  as 
12  + 12 ;  so  that  when  they  come  to  15  X  2,  they  will  have  no 
difficulty  in  deducing  the  rule  for  writing  0  and  carrying  1  when 
they  multiply  the  5  by  2. 

98.  Give  the  pupils  time  to  find  for  themselves  the  quotient 
of  30-^2.     If  it  becomes  necessary  to  show  some  of  them  how  to 
work  the  example,  do  not  elaborate  the  meaning  of  the  1  (ten) 
remainder  when  the  tens'  figure,  3,  is  divided  by  2.     An  experi- 
enced mathematician,  in  dividing  9752  by  2,  does  not  say  2  into 
9  thousand  4  thousand  times  with  a  remainder  of  1  thousand, 
2  into  1700  8  hundred -times  with  a  remainder  of  1  hundred,  etc. 

In  dividing  30  by  2,  children  should  not  be  permitted  to  write 
the  first  remainder,  1,  before  the  0,  to  indicate  that  2  is  to  be 
divided  into  10  for  the  second  quotient  figure.  Children  learn 
to  work  just  as  well  without  these  unnecessary  scaffolds. 

104.  While  these  drill  exercises  introduce  a  multiplier  greater 
than  2,  they  contain  no  combinations,  except  3x3,  other  than 
those  found  in  the  preceding  work.  After  working  these  exam- 
ples, the  pupils  will  have  learned  that  twice  9  is  equal  to  9  twos, — 
that  when  he  knows  the  table  of  2's,  he  knows  a  portion  at  least 
of  the  table  of  3's,  4's,  etc.,  to  9's. 


24  MANUAL  FOR  TEACHERS 

111.  When  the  teacher  places  the  pointer  on  a  number  in  one 
of  the  two  outer  spaces  of  the  first  circle,  the  pupil  promptly 
gives  the  result  obtained  by  adding  to  it  the  number  contained 
in  the  inmost  space.     When  this  last  number  has  been  combined 
with  all  the  others,  it  is  replaced  by  a  different  number. 

112.  These  drills  are  useful  to  impress  upon  a  child  the  fact 
that  when  he  knows,  for  instance,  that  6  and  5  are  11,  he  should 
also  know  that  6  and  15  are  21,  that  6  and  25  are  31,  etc.    They 
may  also  be  employed  as  subtraction  drills. 

115.  Division  drills  are  necessary  to  enable  pupils  to  acquire 
facility  in  obtaining  quotients  and  remainders.  When  pupils 
are  dividing  by  2,  the  numbers  from,  say,  9  to  19  are  written  on 
the  board  with  2  underneath. 

9,   10,    11,   12,    13,   14,   15,   16,   17,   18,   19 


When  the  pointer  is  placed  at  the  9,  the  pupil  answers  4  and 
1;  when  placed  at  14,  he  answers  7;  at  17,  8  and  1;  etc. 
Other  divisors  may  be  employed,  but  care  should  be  taken 
not  to  have  any  quotient  figure  but  1  or  2  at  this  time,  as  pupils 
have  not  yet  learned  the  table  of  3's.  Thus,  when  6  is  used  as 
a  divisor,  the  teacher  should  not  use  a  dividend  greater  than  17. 
When  the  three-times  table  is  known,  numbers  from  12  to  29 
may  be  written. 

Facility  in  division  will  come  only  by  practice,  and  it  may  be 
necessary  for  the  teacher  to  supplement  the  examples  of  the 
book  by  others  of  her  own. 

118.  Do  not  fail  to  keep  up  practice  in  addition  and  sub- 
traction. 

119.  Subtraction  examples  in  which  the  subtrahend  is  given 
before  the  minuend  should  occasionally  be  used. 


NOTES  ON   CHAPTER  TWO  25 

121.  Do  not  worry  a  pupil  by  attempting  to  explain,  through 
problem  9,  the  difference  between  division  and  partition.  Let  him 

write    ^ —  without  taking  advantage  of  the  opportunity 

25  pounds 

to  show  him  that  he  should  have  an  abstract  quotient  when  the 
divisor  and  dividend  are  both  concrete. 

140.  The  analysis  of  problem  3  should  not  be  required.  A 
pupil  that  obtains  from  problem  2  the  knowledge  that  18  five- 
dollar  bills  amount  to  $90  will  probably  get  the  correct  answer 
to  the  next  problem,  even  though  he  may  have  to  use  5  as  a 
multiplier  instead  of  the  18  that  the  more  common  form  of  the 
analysis  would  require.  The  other  form  should  not  be  presented 
at  this  stage. 

143.  Children  should  be  permitted  to  determine  for  themselves 
the  method  of  obtaining  the  half  of  36.  It  may  require  a  little 
longer  time  than  to  show  them,  but  the  time  will  not  be  wasted. 

147.  Roman  notation  is  not  of  much  importance.  Most  chil- 
dren learn  sufficient  about  it  from  the  numbers  affixed  to  their 
reading  lessons. 

157.  Teachers  should  not  endeavor  to  show  by  drawings  that 
a  quart  measure  is  twice  as  large  as  a  pint.  If  a  pint  measure  is 
represented  by  a  rectangle,  each  side  of  the  rectangle  indicating 
the  quart  should  be  only  about  1£  times  that  of  the  former  in 
order  to  preserve  the  correct  ratio,  and  children  are  not  mathe- 
maticians enough  to  understand  that  where  one  of  two  similar 
solids  has  its  corresponding  dimensions  1-J-  times  those  of  the 
other,  the  volume  of  the  former  is  double  that  of  the  latter. 
Use  the  measures  themselves,  borrowing  them,  if  necessary,  from 
a  neighboring  store. 

159.  A  few  problems  involving  more  than  one  operation  are 
here  introduced.  Avoid,  if  possible,  giving  help;  and  do  not 


26  MANUAL   FOR  TEACHERS 

require  the  scholars  to  perform  unnecessary  work,  or  to  follow 
the  same  mode  of  solution  or  arrangement.  In  solving  the  first, 
some  may  write  on  their  slates  only  two  numbers,  viz.  15  and  35. 
Others  may  set  down  15,  15,  and  20,  etc.  Do  not  teach  yet 
how  to  multiply  by  a  mixed  number. 


VI 

NOTES  ON  CHAPTER  THREE 

While  the  teaching  of  formal  definitions  should  find  no  place 
in  the  arithmetical  instruction  of  the  earlier  years,  the  teacher 
should  not  hesitate  to  employ  such  technical  terms  as  are  called 
for  by  the  work  of  the  grade.  Pupils  gradually  learn  to  under- 
stand what  is  meant  by  multiplier,  quotient,  remainder,  etc., 
even  where  no  attempt  is  made  to  explain  the  signification  of  the 
words.  They  will  also  become  able  to  use  each  correctly,  even  if 
they  cannot  state  its  exact  meaning  in  language  that  will  satisfy 
a  critical  mathematician. 

164.  Sight  exercises  in  division  should  be  extended  to  cover 
dividends  that  are  not  multiples  of  the  divisor.  The  slate  exam- 
ples in  division  supplied  thus  far  have  no  remainders,  as  children 
find  it  more  agreeable  in  the  earlier  stages  of  this  work  to  have 
the  answer  a  whole  number.  The  partial  dividends,  however, 
do  not  always  exactly  contain  the  divisor,  hence  the  need  of 
such  drills  as  will  enable  the  pupil  to  determine  rapidly  the 
quotient  figure  and  the  remainder.  Until  Art.  176  is  reached, 
this  remainder  need  not  be  given  by  the  pupils  in  the  form  of  a 
fraction.  See  Art.  115. 

168.  In  making  "  original  problems,"  the  pupil  should  strive 
to  be  original.  No  problem  should  be  accepted  as  satisfactory 
that  is  substantially  the  same  as  one  already  furnished  by 
another  pupil.  If,  for  example,  the  following  is  given  to  illus- 
trate 12  X  5  :  "  What  will  be  the  cost  of  5  yards  of  ribbon  at  12 
cents  a  yard? "  the  teacher  should  not  be  satisfied  with  —  "  How 

27 


28  MANUAL   FOR  TEACHERS 

much  will  be  paid  for  5  pounds  of  cheese  at  the  rate  of  12  cents 
per  pound?" 

174.  While  the  problems  are  gradually  becoming  more  diffi- 
cult, some  of  them  can  be  done  by  bright  pupils  without  using 
the  pencil.  In  these  cases,  require  that  only  the  answers  should 
be  written.  See  previous  notes  on  problem  work.  (Arts.  23, 
84,  and  159.) 

178.  Children  should  be  permitted  to  follow  their  own  26 
plan  of  finding  the  product  of  26  by  1^-.  Some  may  do  +13 
the  work  by  simply  placing  13  under  26.  The  regular  39 
method  should  not  be  taught  until,  perhaps,  the  25th 
example,  as  the  previous  ones  can  be  done  by  the  children  with- 
out assistance.  At  this  point,  however,  the  systematic 
124  way  of  multiplying  by  a  mixed  number  may  be  pre- 

2|      sented,  which  should  be  followed  in  such  subsequent 

31         examples  as  are  not  so  simple  as  to  make  this  amount 
248         of  writing  unnecessary,  as  is  the  case  in  the  26th. 
279  In  finding  £  of  124,  the  pupil  should  not  be  permitted 

to  write  the  multiplicand,  124,  in  some  other  part  of  his 
slate,  and  4  as  a  divisor  in  front  of  it.  No  other  writing  of 
figures  should  be  allowed  than  is  given  above.  A  little  practice 
will  enable  scholars  to  perform  this  division  and  other  similar 
operations,  without  always  bringing  into  close  contact  the  num- 
bers to  be  handled. 

In  some  European  countries,  the  multiplier  is 
760  X  1-J-  placed  at  the  right  of  the  multiplicand,  instead  of 
152  being  written  underneath.  An  example  like  the 

912  26th  would  be  worked  in  that  case  without  writing 

760  a  second  time.     To  small  children,  how- 
ever, it  would  be  confusing  to  be  required  to  learn  two      760 
methods  of  working  examples  so  nearly  alike  ;  hence  the          1^ 
advisability  of  uniformly  following  the  plan  originally      152 
given,  of  first  finding  the  fractional  part,  and  then  multi-      760 
plying  by  the  whole  number.  912 


NOTES  ON  CHAPTER  THREE  29 

180.   The  arrangement  of  work  should  begin  to  receive 

some  attention.     In  solving  the  second  problem,  some  2)70 

children  will  find  the  cost  of  -^  pound  of  tea  on  one  por-  35 

tion  of  the  slate,  and  then  write  this  amount,  35^,  on  -J-25 

another  part,  with  25^  underneath.      They  should  be  60^ 
led  to  see  how  to  avoid  doing  unnecessary  work. 

186.  Some  short  examples  in  the  addition  and  the  subtraction 
of  horizontal  columns  are  given,  to  accustom  children  to  handle 
numbers  that  are  not  arranged  for  work  in  the  usual  way. 
The  addition  example  could  be  used  to  explain  the  reason 
for  "  carrying,"  but  the  explanation  should  be  deferred  for  the 
present. 

191.  Examples  in  division  should  occasionally  be  presented 
to  pupils  in  the  form  used  in  the  second  column.  When  children 
recognize  *f-  as  an  example  in  division,  they  need  no  rule  for 
the  reduction  of  an  improper  fraction  to  a  whole  or  to  a  mixed 
number. 

197.  Do  not  furnish  the  pupils  with  a  method  of  solving  the 
9th  example  that  is  suited  to  a  sixth  year  class  in  denominate 
numbers.  Leave  them  to  their  own  resources  as  much  as  pos- 
sible. 

202.  More  drill  examples  are  needed  than  are  furnished  in 
the  book. 

203.  To  secure  good  work  in  division,  much  practice   must 
be  given.     Many  more  examples  than  are   here  supplied  may 
be  needed  by  some  classes. 

213.  While  it  is  convenient  to  write  the  subtrahend  under 
the  minuend,  pupils  should  gradually  accustom  themselves  to 
perform  the  fundamental  operations  with  numbers  in  other  than 
the  usual  positions. 


30  MANUAL   FOR  TEACHERS 

215.  Children  should  be   encouraged  to   avoid   unnecessary 

7U19  wr^ing.     They  should  be  led  to  see  that  after  finding 

Ty  on  one  part  of  the  slate  that  -f  of  119  is  17,  they  should 

X  5  not  place  this  number  in  another  place  in  order  to  multi- 

"etcT  ply  by  5. 

220-223.  These  drills  are  intended  to  lead  up  to  the  use 
of  larger  numbers  in  the  oral  work  of  the  pupils. 

224.  It  is  not  advisable  to  begin  formal  instructions  in  frac- 
tions at  this  stage  of  school  life.  There  is  no  need  of  defining 
the  word  "fraction"  for  the  present.  Eveiy  member  of  the 
class  will  be  able  to  tell  what  is  the  sum  of  £  and  -£,  especially 
if  the  question  is  put  in  the  form  of  a  problem. 

It  will  be  necessary,  perhaps,  to  explain  that  4  X  %  is  another 
way  of  expressing  -J-  of  4 ;  that  ^-X  10  means  10  halves.  1-^-J- 
will  also  require  translation  into  the  form,  "  How  many  halves 
in  1  ?  "  Pupils  may  be  led  to  see  this  by  being  asked  to  indicate 
by  signs  and  figures  the  example,  "  How  many  twos  are  there 
in  eighteen?"  The  drills  in  the  use  of  fractional  divisors  need 
not  be  made  prominent  for  the  present. 

230.  Accustom  children  to  writing  the  decimal  point  in  the 
product,  as  soon  as  it  is  reached  in  multiplying.     Reasons  should 
not  be  dwelt  upon. 

231.  The  above  applies  to  placing  the  decimal  point  in  the 
quotient. 

238.  Unless  pupils  have  been  carefully  trained  to  give  only 
reasonable  answers  to  slate  problems,  there  will  be  some  who 
will  obtain  171  as  the  sum  of  13  J  and  4J.  They  will  first  write 
1  as  the  equivalent  of  ^  +  ^-;  and  to  this  they  will  prefix  17, 
obtained  by  adding  13  and  4.  The  special  training  in  number 
received  by  pupils  taught  by  the  Grube  method  prevents  to 
a  great  extent  the  absurd  mistakes  found  in  the  answers 


NOTES   ON    CHAPTER    THREE  31 

given  JDV  pupils,  even  of  high-school  classes,  to  simple  problems. 
When  the  early  arithmetical  instruction  is  largely  given  to  work 
in  the  fundamental  processes,  the  teacher  should  make  liberal 
use  of  oral  problems,  to  give  the  requisite  knowledge  of  number 
that  will  enable  a  pupil  to  know  when  his  answer  is  very  much 
out  of  the  way.  Systematic  instruction  in  finding  "  approxi- 
mate "  results  is  supplied  in  later  chapters. 

239.  These  examples  are  intended  to  lead  up  to  finding  the 
difference  between  a  whole  number  and  a  mixed  number. 

240.  Pupils  will   find   little  difficulty  in  working   out  these 
examples  if  they  are  left  to  themselves. 

241.  When  the  addition  and  the  subtraction  of  mixed  num- 
bers containing  halves  are  readily  performed,  the  teacher  will  find 
comparatively  little  trouble  with  the  work  under  Arts.  241-245. 
Encourage  pupils  to  make  diagrams;  or,  if  necessary,  to  divide 
circles  into  quarters,  and  to  use  these  parts  in  performing  the 
required  operations  with  the  fractions. 

To  find,  for  instance,  the  sum  of  £  +  £,  it  may  be  advisable 
to  permit  some  scholars  to  arrange  the  six  quarter-circles  in  such 
a  way  as  to  make  a  whole  circle  and  a  half-circle. 

246.  As  children  are  more  accustomed  to  dealing  with  halves 
and  quarters  than  with  thirds,  a  little  more  illustrative  work 
may  be  needed  in  Arts.  246-250,  than  was  required  in  the  pre- 
vious work  in  the  addition  and  the  subtraction  of  mixed  num- 
bers. 


VII 

NOTES  ON  CHAPTER  FOUR 

253-258.  In  the  last  chapter,  pupils  were  required  to  add 
only  fractions  containing  the  same  denominator  ;  in  this  chapter, 
an  addition  or  a  subtraction  example  may  contain  fractions 
whose  denominators  are  different.  For  the  present,  however, 
it  will  not  be  necessary  to  call  attention  to  the  need  of  reducing 
fractions  to  a  common  denominator.  The  average  scholar  can 
solve  these  examples  without  assistance,  if  he  has  been  able  to 
work  out  those  found  in  Chapter  III. 

259.  While  these  problems  are  becoming  more  difficult,  they 
are  still  well  within  the  powers  of  a  pupil  that  is  really  anxious  to 
solve  them.  When,  however,  they  are  found  to  be  beyond  the 
capacity  of  many  members  of  the  class,  the  teacher  may  first 
use  them  as  "  sight "  problems,  with  some  slight  changes  in  the 
figures. 

If,  for  instance,  after  a  pupil  that  reads  the  first  from  his  book 
declares  that  he  is  unable  to  obtain  the  answer  mentally,  the 
teacher  may  give  it  as  follows : 

A  sailor  has  10  yards  of  cloth.  He  uses  4  yards  for  a  coat 
and  2  yards  for  a  vest.  How  many  yards  has  he  left  ? 

In  the  second,  1|-  pounds  may  be  substituted  for  1^  pounds; 
in  the  third,  3  packages  instead  of  4;  20  dozen  in  the  fourth, 
instead  of  3£  dozen. 

Slate  work  on  these  problems  should  not  be  permitted  until 
so  many  have  been  solved  in  this  way  that  the  pupil  has  had  time 
to  forget  what  operations  have  been  used  in  each.  This  will 

32 


NOTES   ON   CHAPTER   FOUR  33 

require  him  to  study  the  conditions  of  the  different  problems, 
instead  of  relying  upon  his  memory. 

266.  When  the  formal  analysis  of  oral  problems  is  made 
a  feature  of  the  work,  it  is  important  that  the  statements  be 
not  so  long  as  to  be  tedious. 

In  the  first,  for  example,  the  following  would  be  sufficient, 
after  the  pupil  has  stated  the  problem: 

"  If  8  ounces  of  tea  cost  40  cents,  1  ounce  will  cost  5  cents, 
and  5  ounces  will  cost  25  cents." 

While  the  customary  order  has  been  followed  in  the  systematic 
treatment  of  the  various  topics,  pupils  are  called  upon  in  the 
earlier  chapters  of  Mathematics  for  Common  /Schools  to  solve 
many  problems  that  are  frequently  deferred  in  other  books  to  a 
later  stage  of  their  arithmetical  instruction.  While  scholars 
readily  solve  this  class  of  problems,  they  are  not  always  able 
to  state  in  technical  language  the  reasons  for  the  various  proc- 
esses employed  in  obtaining  the  answers.  A  child  who  sees  that 
division  is  used  to  ascertain  the  number  of  ten-cent  pies  that  can 
be  purchased  for  forty  cents,  cannot  be  made  to  understand  thus 
early  in  his  school  life  that  the  same  process  is  used  to  find  what 
part  of  such  a  pie  can  be  bought  for  five  cents.  A  correct  state- 
ment by  the  pupil  of  his  method  of  reaching  the  result,  should 
usually  be  accepted  as  satisfactory.  Even  in  the  more  simple 
questions,  set  forms  of  analysis  should  be  carefully  avoided. 

268.  To   prevent   misunderstanding,   parentheses   have   been 
employed  even  when  not  required  by  arithmetical  usage.     The 
quantities  within  the   parentheses  must  be  added,  multiplied, 
etc.,  before  being  operated  upon  by  the  quantity  outside.     The 
third    example  becomes   30  X  3  ;    the  fourth,  80  -4-  4 ;    the  fifth, 
\  of  80;  the  eighth,  70 -f-  7,  etc. 

269.  These  may  be  used  as  slate  examples,  if  they  are  found 

too  difficult  for  "  sight  "  work. 


34  MANUAL   FOR   TEACHERS 

271.  Some  of  these  questions  may  not  require  the  use  of  a 
pencil ;  Nos.  6,  7,  8,  11,  and  19,  for  instance. 

272.  The  answers  to  the  first  ten  examples  should  be  given  at 
sight. 

273.  Use  49  to  57,  inclusive,  as  "  sight "  examples ;  also  as 
many  as  possible  of  those  in  the  next  section. 

274.  When  the  divisor  ends  in  one  or   more 
ciphers,  the  latter  are  set  off  by  a  vertical  bar, 
and  also  a  corresponding  number  of  figures  from 

the  right  of  the  dividend.     To  keep  the  pupil  from  omitting 

these  figures  from  the  remainder,  it  is  advisable  to  require  him 

to  write  the  partial  remainder  as  above,  before  he 

8 1 0)434 [1         begins  to  divide.     Then,  using  8  as  a  divisor,  he 

54|^         writes  the  quotient  figures  in  their  places,  and 

completes  the  partial  remainder  by  prefixing  2  to 

the  1  that  was  originally  brought  down. 

It  being  the  usual  practice  in  abstract  examples        8|0)434!0 
in  division  to  refrain  from  reducing  the  fractional  54ff 

part  of  the  quotient  to  lowest  terms,  the  above 

method  may  be  used  in  examples  where  both  the 
80)4340         divisor  and  the   dividend  terminate   in   a   cipher. 
54f         Some  teachers  prefer,  however,  in  this  case,  to  can- 
cel the  cipher  in  each,  and  to^  give  the  quotient 
of  4340  -*-  80  as  54f . 

277.  Employ  in  "  sight "  work. 

278.  The  foot-rule  and  the  yardstick  should  be  used  by  the 
children.      They  should  ascertain,  for  instance,  the  length  of 
their  slates  in  inches,  the  length  of  the  blackboard  in  yards  or 
in  feet,  the  height  of  the  blackboard  in  feet,  the  dimensions 
of  the  room,  etc. 


NOTES   ON   CHAPTER   FOUR  35 

280.  It  will  be  sufficient  to  accustom  pupils  to  placing  the 
product  by  the  tens'  figure  one  place  to  the  left  without  giving 
the  reason  therefor.  Neatness  in  the  arrangement  of  the  work, 
and  the  careful  writing  of  figures,  will  prevent  some  mistakes. 

282.    In  short  division,  the  scholar  has  been  taught  to  place 
the  first  figure  of  the  quotient  under  the  last  figure 
of  its   partial   dividend,    and    to    write   under   each       — ^ 
succeeding  figure  of  the  dividend    its  corresponding 
quotient  figure.     When  his  work  is  neatly  arranged,  he  seldom 
omits  ciphers,  nor  does  he  often  obtain  two  quotient  figures  from 
one  partial  dividend. 

To  obtain  the  benefit  of  this  experience,  the  pupil  should 
be  taught  in  long  division  to  write  the  quotient 
over  the  dividend.  By  doing  this,  he  will  not  be 
tempted,  as  are  some  beginners  that  place  the 
quotient  at  the  right,  to  give  23-2lr  as  the  answer 
to  the  above  example ;  nor  will  he  be  likely  to 
think  that  252  contains  21,  111  times.  This  last 
result  is  obtained  by  assuming  that  the  second 
partial  dividend,  42,  contains  the  divisor  1  time,  with  a  re- 
mainder of  21.  This  latter  is  then  made  a  partial  dividend, 
with  the  above  result. 

285.   While  the   pupil   may  write  16  as  a  multiplier  in  the 
5th  problem,  he  should  be  required  to  multiply  by  80,         ^ 
in  order  to  shorten  the  work.     The  multiplication  by  30         -,„ 
should  be  performed,  also,  without  rewriting  the  numbers     — 
so  as  to  place  30  under  16. 

286-290.  The  special  drills  will  be  found  of  great  value  in 
giving  pupils  a  knowledge  of  numbers ;  and  many  oral  problems 
employing  these  and  similar  combinations  should  be  made  by 
the  teacher.  Oral  problems  containing  large  numbers  should, 
as  a  rule,  require  but  one  operation  for  their  solution. 


36  MANUAL   FOR   TEACHERS 

In  the  oral  addition  of  numbers  of  two  figures,  the  pupil 
should  not  commence,  as  in  slate  work,  with  the  units'  figures. 
The  special  drills  of  the  last  chapter  should  have  taught  him  to 
think  immediately  of  80  when  he  sees  40  +  40,  60  +  20,  50  +  30, 
etc.  The  next  step  in  this  work  should  contain  such  combina- 
tions as  47  +  40,  63  +  20,  54  +  30,  etc.  In  adding  54  and  30, 
the  pupil  should  be  taught  to  first  see  the  eighty,  then  the  four. 
The  sum  of  27  and  32  (the  third  step)  should  be  obtained  by 
joining  27  and  30  to  make  57,  and  adding  2  to  this  result  to 
obtain  59.  If  the  pupil  begins  with  the  units,  7  and  2,  he  is 
likely  to  forget  the  tens'  figures.  When  the  addition  work 
is  readily  performed,  the  pupil  finds  little  trouble  with  the  rest. 
There  being  no  carrying,  he  will  readily  obtain  the  product  of 
32  by  3,  and  the  others  given  in  Art.  288,  especially  if  he  begins 
the  multiplication  at  the  tens'  figure.  After  he  becomes  expert 
in  adding  and  multiplying,  he  will  experience  no  difficulty  in 
subtracting  and  dividing. 

294.    The   teacher   should   not   encourage  unnecessary  work, 

by  permitting  children  to  write  the  sum  of  12J  +  6J  as  18|  = 

18  +  1  =  19.      If,  however,  it  be  deemed   advisable  in  the  4th 

example,  for  instance,  that  the    fractions   should   be    expressed 

with  the   same   denominator,  care   must   be   taken   to   prevent 

pupils  fr°m  making  such  mistakes  as  using  the  sign  of 

equality  between  ^  and  J-  in  such  a  way  as  to  represent 

— i-l-4     that  50J  is  the  equivalent  of  J-.     A  vertical  line  drawn 

5  ¥  I  ¥     between  the  two  sets  of  fractions  will  serve  to  separate 

the  original  example  from  the  auxiliary  portion.    (See  Art.  310.) 

304.  As  some  children  merely  look  for  the  figures  of  a  prob- 
lem without  paying  attention  to  its  terms,  an  occasional  one 
is  given  in  which  some  or  all  of  the  numbers  are  expressed 
in  words. 

306.  In  making  out  a  bill,  it  is  convenient  to  be  able  to  write 
the  cost  of  196  Ib.  at  4^  per  Ib.  without  using  another  sheet  of 


NOTES  ON   CHAPTER   FOUR  37 

paper  and  placing  the  4  under  196.  In  working  these  examples, 
the  pupil  is  expected  to  write  only  one  figure  of  the  product  at  a 
time.  It  is  not  intended  that  all  of  these  twenty-five  examples 
should  be  done  before  proceeding  with  the  subsequent  work. 
A  few  of  them  should  be  used  from  time  to  time  throughout 
the  term. 

307.  The  last  sentence  applies  also  to  these  examples.  A  few 
of  the  easier  ones  should  first  be  given.  After  more  practice 
in  long  division,  the  more  difficult  ones  may  be  taken  up. 

310.   Whenever  it  becomes  necessary,  in  the  8 

opinion  of  the  teacher,  to  permit  the  rewriting       49J    7 
of  the  fractions  with  a   common    denominator,       20|-   4 
she  should,  as  soon  as  possible,  have  her  pupils       70f  ^  =  If 
write   the   common    denominator  only  once,  as 

above.      When   the   common   denominator  is  written 
49|-   J        under  each  numerator,  it  is  likely  to  be  confusing  to 
20^-  f       -children,  not  to  speak  of  the  danger  of  its  being  added 
-^       in  occasionally  with  the  numerators. 

312.   See  Art.  268. 

316.  Where  the  multiplier  ends  with  ciphers,  some 
teachers  think  that  time  is  saved  by  omitting  the 
ciphers  from  the  partial  products.  The  ciphers  at  the 
right  of  the  multiplier  are  written  beyond  the  multipli- 
cand, and  are  brought  down  at  the  end  of  the  work.  98800 

Other  teachers  prefer  to  place  the  numbers  as  is  gen- 
erally done  in  multiplication,  writing  a  cipher  under  each  one 
in  the  multiplicand  as  its  partial  product,  and  writing 
76       the  partial  products  by  3  and  1  under  these  figures, 
1300      respectively.     This  method  will  be  found  to  give  more 
22800      satisfactory  results  later  on,  when   pupils   have  such 

76 multipliers  as  20£,  300J,  etc.,  in  which  a  fraction  fol- 

98800      lows  the  ciphers. 


38  MANUAL   FOR  TEACHERS 

319.   See  Art.  310.     When  an  addition  example  consists  of 
more  than  two  mixed  numbers  with  fractions  of  different  denom- 
inators, it  may  be  advisable  to  permit  young 
children  to  write  out  the  successive  opera- 
tions in  the  manner  here  indicated. 


7* 

Ji 

22J 


Many  of  the  fifty  examples  on  this  page 
should  be   used  as  "sight"  work  from   the 


=  If  =  1%      blackboard,    the    pupils    writing    only    the 
results.     Nos.   1-6,  8-9,  13-16,   23-24,   26, 
31-37,  41-43,  49-50  can  be  treated  in  this  way  after  they  have 
been  worked  out  on  the  slate,  if  not  in  the  first  instance. 

321.  Until  children  obtain  some  knowledge  of  numbers,  their 
progress  in  long  division  is  very  slow.  In  dividing  918  by  17, 
for  instance,  a  pupil  that  is  not  properly  instructed  will  some- 
times take  1  as  the  first  figure  of  the  quotient.  When,  after 
subtracting,  he  obtains  a  remainder  of  74,  he  may  realize  that 
he  is  wrong  without  being  able  to  determine  just  how  far  astray 
he  is.  In  this  case  he  tries  2  as  the  quotient  figure,  ascertaining 
the  product  of  17x2  in  a  corner  of  his  slate,  and  then  trans- 
ferring the  34  to  its  proper  position  under  the  first  two  figures 
of  the  dividend.  Another  subtraction  follows,  with  a  resulting 
remainder,  again,  perhaps,  recognized  as  too  great ;  and  so  on. 

The  object  of  these  drills  is  to  enable  the  scholar  to  reach  at 
once  a  close  approximation  to  the  correct  quotient  figure.  Their 
use  may  be  commenced  in  some  such  way  as  the  following : 

The  teacher  writes  on  the  blackboard  a  convenient  number  of 
those  found  among  the  first  twenty,  arranging  them  as  shown 
below,  with  the  divisor  preceding  the  dividend.  Under  these 
she  places  the  corresponding  ones  from  the  second  and  third 
sets,  respectively. 

20)160  60)360  90)450  50)300  30)270 
19)160  59)360  89)450  49)300  29)270 
21)160  61)360  91)450  51)300  31)270 


NOTES  ON   CHAPTER   FOUR  39 

Placing  the  pointer  on  those  in  the  first  row,  successively,  she 
receives  the  quotients  promptly.  She  then  asks  for  the  quotient 
of  the  first  in  the  third  row,  21)160.  If  the  pupil  announces 
8  as  the  result,  he  should  be  required  to  give  mentally  the  prod- 
uct of  21  X  8,  which  he  will  find  to  be  too  great.  He  is  thus 
led  to  see  that  the  quotient  is  7,  with  a  remainder.  The  other 
quotients  in  this  row  are  then  elicited.  After  a  pupil  discovers 
that  21  is  not  contained  8  times  in  160,  that  61  is  not  con- 
tained 6  times  in  360,  etc.,  he  may  be  introduced  to  the  second 
row.  A  little  questioning  will  enable  him  to  perceive  that  if 
160  -5-  20  =  8,  the  quotient  of  160  •+•  19  must  be  at  least  8,  with 
a  remainder;  that  360-^59  gives  a  quotient  of  6,  with  something 
over,  etc.  Regular  practice  with  this  particular  set  of  drills 
will  rob  division  by  19,  29,  39,  etc.,  of  some  of  its  terrors  to 
slow  pupils,  as  they  will  be  led  to  use  2,  3,  4,  etc.,  as  "  trial 
divisors"  instead  of  1,  2,  3,  etc.,  whereby  they  will  be  able  to 
obtain  their  answers  in  a  reasonable  time. 

After  the  children  have  become  able  to  announce  at  once  the 
quotients  of  all  the  drills  in  the  first  three  sets,  and  other  similar 
ones  supplied  by  the  teacher,  they  may  take  up  the  remaining 
ones  by  degrees.  When  there  is  a  remainder,  the  pupils  should 
not  be  required  to  calculate  it. 

324.  The  quotient  of  2,800-^200  may  be  made  more  obvious 
if  the  dividend  is  read  28  hundred,  instead  of  two  thousand  eight 
hundred. 

328.  See  Art.  274,  as  to  writing  the  partial  remainder  before 
beginning  to  divide. 

341.  Do  not  give  reasons  for  the  location  of  the  partial  prod- 
ucts. There  is  plenty  of  time  for  the  science  of  arithmetic  later 
on  in  the  course. 

343.  Although  the  divisors  contain  three  or  four  figures,  these 
examples  should  not  prove  so  difficult  as  many  of  those  already 


40  MANUAL   FOR   TEACHERS 

worked.     A  pupil  that  has  learned  from  the  pre- 
i_     vious  drills  that  800 -5- 200-=  4,  will  be  able  to  see 
201)8643     that  201  is  contained  4  times  in  the  first  three  fig- 
ures  of  8,643.     The  teacher  should  be  careful  to 

see  that  the  first  quotient  figure  is  written  in  its 
etc.  , 

proper  place. 

No.  36  may  cause  some  hesitation  until  the  pupil  perceives  that 
he  has  to  divide  81  hundred  and  something  by  9  hundred  and 
something.  No.  37  will  become  simple  if  handled  in  the  same 
way.  In  No.  47,  98  hundred  divided  by  12  hundred  will  give 
the  clue  to  the  quotient ;  in  Nos.  48  and  50,  nine  thousand  and 
two  thousand  should  be  used  for  this  purpose. 

344.  With  such  a  multiplier  as  209,  some  teachers 
write  a  series  of  ciphers  to  denote  the  product  by  0. 
The  method  given  in  the  text-book  is  the  one  gener- 
ally followed  in  later  school  life,  and  is  just  as  easily 
taught  to  beginners  as  the  above. 

346.   Where  the  multiplicands  are  small,  as  in  nearly  all  of 
these  examples,  the   product   by  the   fraction   should   be 
determined  "  mentally"  and  written  in  its  place.     A  pupil         3 
should  not  be  encouraged  to  waste  time  by  indicating  on     — E 

another  part  of  his  slate  that  64  is  to  be  divided 
^—     by  8,  and  that  this  product  is  to  be  multiplied  by 
3,  and  doing  all  this  work  to  reach  a  result  that 
— -     can  be  readily  obtained  without  any  writing  whatever. 
In  Nos.  78  and  88,  such  pupils  as  need  to  use  the  pencil 
in  multiplying  by  the  fraction    should    be  permitted  to  do  so. 
The  teaching  of  the  common  method  of  multiplying  by  a  mixed 
number  is  taken  up  at  the  beginning  of  the  next  chapter. 


VIII 

NOTES  ON  CHAPTER  FIVE 

347.  The  denominators  of  fractional  multipliers  have  hereto- 
fore been  factors  of  the  multiplicands,  and  the  latter  have  been, 
as  a  rule,  small  numbers.     With  the  introduction  of  larger  num- 
bers and  the  occasional  use  of  multiplicands  that  are  not  multiples 
of  the  denominators  of  the  fractions  in  the  multiplier,  it  becomes 
necessary  to  furnish  pupils  with  a  general  method  of  dealing  with 
this  class  of  examples.     (See  Arithmetic,  Art.  347.) 

348.  In  multiplying  27  by  13^,  some  pupils  may 

be  tempted  to  follow  the  rule,  and  to  multiply  27  by  -.^ 

the  numerator  1.     In  the  first  few  examples  this  may  q\  07^ 

be  permitted,  but  the  scholars  should  soon  be  taught  — Q- 

to  discontinue  the  practice,  and  to  divide  the  multipli-  , 
cand  without  rewriting  it.     (See  Art.  178.) 

350.  In  adding  56  and  17,  the  pupil  should  first  combine  56 
and  10  to  make  66,  and  then  add  7.     (See  Art.  286.) 

351.  Children  taught  subtraction  by  the  "  building-up  "  method 
will  ascertain  how  many  must  be  added  to  19  to  make  66,  by 
saying  19  and  40  are  59,  and  7  are  66 ;  or  19  and  7  are  26,  and 
40  are  66.     While  the  second  plan  is  easier  in  some  respects,  it 
gives  the  40  and  the  7  of  the  result  in  the  reverse  order,  which 
makes  it  necessary  for  the  pupils  to  transpose  them.     In  this 
respect,  the  first  plan  is  more  satisfactory. 

41 


42  MANUAL   FOR   TEACHERS 

When  the  other  method  of  subtraction  is  practiced  in  slate 
work,  66  is  first  diminished  by  10  and  then  by  9.  To  find  the 
difference  between  94  and  76,  the  pupil  takes  70  from  94,  leaving 
24,  and  from  this  remainder  takes  6. 

352.  In  multiplying  24  by  4  the  pupil  begins  at  the  tens. 
Four  times  20  are  80,  to  which  is  added  4x4,  making  96. 

353.  While  nearly  the  whole  class  will  learn  to  give  answers 
mentally  to  the  previous  combinations,  it  may  be  necessary  to 
use  the  division  drills  as  "  sight "  work  chiefly. 

359.   See  Art.  319. 

362.  Oral  problems  involving  several  operations,  or  those  of 
an  unfamiliar  type,  should  be  solved  from  the  book  as  "sight" 
work,  and  should  be  followed  later  on  by  similar  questions  an- 
swered without  seeing  the  numbers.    No.  5  is  of  the  second  kind ; 
and  it  might  be  well  to  place  it  on  the  board,  writing  "  2  thirds  '* 
and  "  1  third  "  to  express  the  parts,  instead  of  employing  the 
fractional  form  or  that  given  in  the  book.     In  No.  7,  the  quotient 
of  60-^40  will  be  expressed  by  1J,  instead  of  the  1|^-  obtained  by 
writing  the  remainder  over  the  divisor.      No.  5  should  not  be 
made  an  excuse  for  teaching  a  method  of  obtaining  the  cost  of 
the  whole  when  that  of  a  part  is  given. 

These  examples  are  introduced  to  give  variety  to  the  work,  to 
lay  a  foundation  for  subsequent  systematic  treatment  of  problems 
of  this  kind,  and  to  give  a  pupil  an  opportunity  to  use  his  think- 
ing powers.  The  way  to  deprive  them  of  value  is  to  "  explain  " 
how  they  should  be  done,  or  to  require  from  the  scholars  too 
much  analysis. 

363.  If  the  school  does  not  own  these  measures,  the  teacher 
should  endeavor  to  secure  the  loan  of  a  quart,  a  peck,  and  a 
bushel,  for  a  few  hours,  at  least.     Sawdust  could  be  used  to  show 
pupils  that  the  peck  contains  eight  quarts,  etc. 


NOTES  ON   CHAPTER   FIVE  43 

364.  While  many  of  the  problems  of  this  article  resemble  the 
previous  oral  problems,  it  may  be  advisable  to  solve  a  number  of 
them  as  "  sight "  work,  changing  the  numbers  when  necessary. 
The  first  may  be  read  "  How  many  200-lb.  barrels  can  be  filled 
from  6,000  Ib.  ?  "  In  the  second  and  third,  the  fractions  may  be 
omitted.  The  cost  of  the  calico  and  of  the  ribbon  in  No.  4  may 
be  made  10  $.  Nos.  5  and  6  need  no  change,  perhaps. 

370-372.  Do  not  waste  time  by  endeavoring  to  use  these  ex- 
amples to  explain  "  carrying  "  or  the  local  value  of  digits. 

374.  The  answers  should  be  written  directly  from  the  book. 
Do  not  permit  scholars  to  copy  the  examples  on  their  slates. 

377.  First,  perform  operations  on  the  quantities  enclosed  within 
the  parentheses. 

384.  Very  little  preliminary  explanation  will  be  needed-.    Place 
f  \         *>     (a)   on  the  blackboard,  and    ask  a     /^\         ? 

125     pupil  to  write  the  missing  number  125 

632     in  its  place,  one  figure  at  a  time,  632 

999     beginning    with    the    units'    figure.  1000 

Have  another  pupil  work  (b)  in  the  same  way.  Nos.  1  to  5  may 
be  used  as  a  class  exercise,  each  pupil  writing  only  the  answer  on 
his  paper,  the  examples  being  placed  on  the  board. 

385.  In  many  German  schools,  children  are  not  permitted  in 
long  division  to  write  the  partial  products.     Examples  6-23  are 
given  to  train  pupils  to  omit  these  products  when  the  quotient 
contains  but  one  figure.     After  a  few  of  them  are  worked  on  the 
board,  the  answers  to  the  others  may  be  written  by  all  the  pupils, 
as  suggested  in  the  preceding  article.     In  writing  the  answers, 
the  pupils  should  first  set  down  the  quotient  figure,  then  the 
divisor  as  the  denominator  of  a  fraction,  and  lastly  the  remainder 
as  a  numerator.     (See  Art.  563,  p.  55.) 


44  MANUAL   FOR  TEACHERS 

386-388.  These  examples  should  be  placed  on  the  board,  and 
the  pupils  should  write  the  results  one  figure  at  a  time. 

397-401.   See  Art.  321. 
405-406.   See  Arts.  306  and  307. 

407.  Prove  the  correctness  of  the  grand  total  by  comparing 
the  total  of  the  6th  column  with  that  of  the  llth  row. 

412.  Permit  the  pupils  to  use  their  own  method  of  working 
these  examples,  and  avoid  giving  unnecessary  assistance. 

413-414.  Example  1  should  be  omitted  where  pupils  do  not 
receive  marks  that  are  thus  averaged.  No.  2  may  also  be  omitted 
if  the  word  "average"  is  not  understood  by  the  pupils. 

424-426.   See  Arts.  286-290,  page  34. 

429.  In  Examples  1,  2,  5,  9,  etc.,  it  will  hardly  be  necessary 
to  inform  the  pupils  that  1  is  not  considered  a  factor  of  a  number. 


SUPPLEMENT 


DEFINITIONS,  PRINCIPLES,   AND  RULES 

A  Unit  is  a  single  thing. 

A  Number  is  a  unit  or  a  collection  of  units. 

The  Unit  of  a  Number  is  one  of  that  number. 

Like  Numbers  are  those  that  express  units  of  the  same  kind. 

Unlike  Numbers  are  those  that  express  units  of  different  kinds. 

A  Concrete  Number  is  one  in  which  the  unit  is  named. 

An  Abstract  Number  is  one  in  which  the  unit  is  not  named. 

Notation  is  expressing  numbers  by  characters. 

Arabic  Notation  is  expressing  numbers  by  figures. 

Eoman  Notation  is  expressing  numbers  by  letters. 

Numeration  is  reading  numbers  expressed  by  characters. 

The  Place  of  a  Figure  is  its  position  in  a  number. 

A  figure  standing  alone,  or  in  the  first  place  at  the  right  of  other 
figures,  expresses  ones,  or  units  of  the  first  order. 

A  figure  in  the  second  place  expresses  tens,  or  units  of  the 
second  order. 

A  figure  in  the  third  place  expresses  hundreds,  or  units  of  the 
third  order ;  and  so  on. 

A  Period  is  a  group  of  three  orders  of  units,  counting  from  right 
to  left. 

RULE  FOR  NOTATION.  —  Begin  at  the  left,  and  write  the  hun- 
dreds, tens,  and  units  of  each  period  in  succession,  filling  vacant 
places  and  periods  with  ciphers. 

i 


11  SUPPLEMENT 

RULE  FOR  NUMERATION.  —  Beginning  at  the  right,  separate  the 
number  into  periods. 

Beginning  at  the  left,  read  the  numbers  in  each  period,  giving 
the  name  of  each  period  except  the  last. 

ADDITION 

Addition  is  finding  a  number  equal  to  two  or  more  given  num- 
bers. 

Addends  are  the  numbers  added. 
The  Snm,  or  Amount,  is  the  number  obtained  by  addition. 

PRINCIPLE.  —  Only  like  numbers,  and  units  of  the  same  order 
can  be  added. 

RULE.  —  Write  the  numbers  so  that  units  of  the  same  order  shall 
le  in  the  same  column. 

Beginning  at  the  right,  add  each  column  separately,  and  write 
Ihe  sum,  if  less  than  ten,  under  the  column  added. 

When  the  sum  of  any  column  exceeds  nine,  write  the  units  only, 
and  add  the  ten  or  tens  to  the  next  column. 

Write  the  entire  sum  of  the  last  column. 

SUBTRACTION 

Subtraction  is  finding  the  difference  between  two  numbers. 
The  Subtrahend  is  the  number  subtracted. 
The  Minnend  is  the  number  from  which  the  subtrahend  is  taken. 
The  Eemainder,  or  Difference,  is  the  number  left  after  subtracting 
one  number  from  another. 

PRINCIPLES.  —  Only  like  numbers  and  units  of  the  same  order 
can  be  subtracted. 

The  sum  of  the  difference  and  the  subtrahend  must  equal  the 
minuend. 

RULES.  — I.  Write  the  subtrahend  under  the  minuend,  placing 
units  of  the  same  order  in  the  same  column. 


DEFINITIONS;    PRINCIPLES,    AND   RULES  ill 

Beginning  at  the  right,  find  the  number  that  must  be  added  to 
the  first  figure  of  the  subtrahend  to  produce  the  figure  in  the  corre- 
sponding order  of  the  minuend,  and  write  it  below.  Proceed  in 
this  way  until  the  difference  is  found. 

If  any  figure  in  the  subtrahend  is  greater  than  the  corresponding 
figure  in  the  minuend,  find  the  number  that  must  be  added  to  the 
former  to  produce  the  latter  increased  by  ten ;  then  add  one  to  the 
next  order  of  the  subtrahend  and  proceed  as  before. 

II.  Beginning  at  the  units'1  column,  subtract  each  figure  of  the 
subtrahend  from  the  corresponding  figure  of  the  minuend  and 
write  the  remainder  below. 

If  any  figure  of  the  subtrahend  is  greater  than  the  corresponding 
figure  in  the  minuend,  add  ten  to  the  latter  and  subtract;  then, 
(a)  add  one  to  the  next  order  of  the  subtrahend  and  proceed  as 
before ;  or,  (b)  subtract  one  from  the  next  order  of  the  minuend 
and  proceed  as  before. 

MULTIPLICATION 

x. 

Multiplication  is  taking  one  number  as  many  times  as  there  are 
units  in  another  number. 

The  Multiplicand  is  the  number  taken  or  multiplied. 

The  Multiplier  is  the  number  that  shows  how  many  times  the 
multiplicand  is  taken. 

The  Product  is  the  result  obtained  by  multiplication. 

PRINCIPLES.  —  The  multiplier  must  be  an  abstract  number. 
The  multiplicand  and  the  product  are  like  numbers. 
The  product  is  the  same  in  whatever  order  the  numbers  are 
multiplied. 

RULE.  —  Write  the  multiplier  under  the  multiplicand,  placing 
units  of  the  same  order  in  the  same  column. 

Beginning  at  the  right,  multiply  the  multiplicand  by  the  number 
of  units  in  each  order  of  the  multiplier  in  succession.  Write  the 


IV  SUPPLEMENT 

figure  of  the  lowest  order  in  each  partial  product  under  the  figure 
of  the  multiplier  that  produces  it.     Add  the  partial  products. 

To  multiply  by  10,  100,  1000,  etc, 

RULE.  —  Annex  as  many  ciphers  to  the  multiplicand  as  there 
are  ciphers  in  the  multiplier. 

DIVISION 

Division  is  finding  how  many  times  one  number  is  contained  in 
another,  or  finding  one  of  the  equal  parts  of  a  number. 
The  Dividend  is  the  number  divided. 
The  Divisor  is  the  number  contained  in  the  dividend. 
The  Quotient  is  the  result  obtained  by  division. 

PRINCIPLES.  —  When  the  divisor  and  the  dividend  are  like  num- 
bers, the  quotient  is  an  abstract  number. 

When  the  divisor  is  an  abstract  number,  the  dividend  and  the 
quotient  are  like  numbei'S. 

The  product  of  the  divisor  and  the  quotient,  plus  the  remainder ', 
if  any,  is  equal  to  the  dividend. 

RULE.  —  Write  the  divisor  at  the  left  of  the  dividend  with  a  line 
between  them. 

Find  how  many  times  the  divisor  is  contained  in  the  fewest  fig- 
ures on  the  left  of  the  dividend,  and  write  the  result  over  the  last 
figure  of  the  partial  dividend.  Multiply  the  divisor  by  this  quotient 
figure,  and  write  the  product  under  the  figures  divided.  Subtract 
the  product  from  the  partial  dividend  used,  and  to  the  remainder 
annex  the  next  figure  of  the  dividend  for  a  new  dividend. 

Divide  as  before  until  all  the  figures  of  the  dividend  have  been 
used. 

If  any  partial  dividend  will  not  contain  the  divisor,  write  a 
cipher  in  the  quotient,  and  annex  the  next  figure  of  the  dividend. 

If  there  is  a  remainder  after  the  last  division,  write  it  after  the 
quotient  with  the  divisor  underneath. 


DEFINITIONS,    PRINCIPLES,    AND   RULES  V 

FACTORING 

An  Exact  Divisor  of  a  number  is  a  number  that  will  divide  it 
without  a  remainder. 

An  Odd  Number  is  one  that  cannot  be  exactly  divided  by  two. 

An  Even  Number  is  one  that  can  be  exactly  divided  by  two. 

The  Factors  of  a  number  are  the  numbers  that  multiplied  to- 
gether produce  that  number. 

A  Prime  Number  is  a  number  that  has  no  factors. 

A  Composite  Number  is  a  number  that  has  factors. 

A  Prime  Factor  is  a  prime  number  used  as  a  factor. 

A  Composite  Factor  is  a  composite  number  used  as  a  factor. 

Factoring  is  separating  a  number  into  its  factors. 

To  find  the  Prime  Factors  of  a  Number. 

RULE.  —  Divide  the  number  by  any  prime  factor.  Divide  the 
quotient,  if  composite,  in  like  manner;  and  so  continue  until  a 
prime  quotient  is  found.  The  several  divisors  and  the  last  quotient 
will  be  the  prime  factors. 

CANCELLATION 

Cancellation  is  rejecting  equal  factors  from  dividend  and  divisor. 
PRINCIPLE.  —  Dividing  dividend  and  divisor  by  the  same  num~ 
ber  does  not  affect  the  quotient. 

GREATEST  COMMON  DIVISOR 

A  Common  Factor  (divisor  or  measure)  is  a  number  that  is  a 
factor  of  each  of  two  or  more  numbers. 

A  Common  Prime  Factor  is  a  prime  number  that  is  a  factor  of 
each  of  two  or  more  numbers. 

The  Greatest  Common  Factor  (divisor  or  measure)  is  the  largest 
number  that  is  a  factor  of  each  of  two  or  more  numbers. 

Numbers  are  prime  to  each  other  when  they  have  no  common 
factor. 


VI  SUPPLEMENT 

The  greatest  common  divisor  of  two  or  more  numbers  is  the 
product  of  their  common  prime  factors. 

PRINCIPLES.  —  A  common  divisor  of  two  numbers  is  a  divisor 
of  their  sum,  and  also  of  their  difference. 

A  divisor  of  a  number  is  a  divisor  of  every  multiple  of  that 
numb e?' ;  and  a  common  divisor  of  two  or  more  numbers  in  a 
divisor  of  any  of  their  multiples. 

To  find  the  Common  I  rime  Factors  of  Two  or  More  Numbers. 

RULE.  —  Divide  th(  numbers  by  any  common  prime  factors, 
and  the  quotients  in  like  manner,  until  they  harp,  no  common 
factor  ;  the  several  divisors  are  the  common  prime  factors. 

To  find  the  Greatest  Common  Divisor  of  Numbers  that  are  Easily 
Factored. 

RULE.  —  Separate  the  numbers  into  their  prime  factors ;  the 
product  of  those  that  are  common  is  the  greatest  common  divisor. 

To  find  the  Greatest  Common  Divisor  of  Numbers  that  are  not 
Easily  Factored. 

RULE.  —  Divide  the  greater  number  by  the  less ;  then  divide 
the  last  divisor  by  the  last  remainder,  continuing  until  there  is  no 
remainder.  The  last  divisor  is  the  greatest  common  divisor. 

2f  there  are  more  than  two  numbers,  find  the  greatest  common 
divisor  of  two  of  them;  then  of  that  divisor  and  another  of  the 
numbers  until  all  of  the  numbers  have  been  used.  The  last  divisor 
is  the  greatest  common  divisor. 

LEAST  COMMON  MULTIPLE 

A  Multiple  of  a  number  is  a  number  that  exactly  contains  that 
number. 

A  Common  Multiple  of  two  or  more  numbers  is  a  number  that 
is  a  multiple  of  each  of  them. 

The  Least  Common  Multiple  of  two  or  more  numbers  is  the 
smallest  number  that  is  a  common  multiple  of  them. 


DEFINITIONS,    PRINCIPLES,    AND   RULES  Vll 

PRINCIPLES.  —  A  multiple  of  a  number  contains  all  the  prime 
factors  of  that  number. 

A  common  multiple  of  two  or  more  numbers  contains  each  of 
the  prime  factors  of  those  numbers. 

The  Least  Common  Multiple  of  two  or  more  numbers  contains 
only  the  prime  factors  of  each  of  the  numbers. 

To  find  the  Least  Common  Multiple  of  Two  or  More  Numbers. 

RULE.  —  Divide  by  any  prime  number  that  is  an  exact  divisor  of 
two  or  'more  of  the  numbers,  and  write  the  quotients  and  undivided 
numbers  below.  Divide  these  numbers  in  like  manner,  continuing 
until  no  two  of  the  remaining  numbers  have  a  common  factor. 
The  product  of  the  divisors  and  remaining  numbers  is  the  least 
common  multiple. 

FRACTIONS 

A  Fraction  is  one  or  more  of  the  equal  parts  of  anything. 

The  Unit  of  a  Traction  is  the  number  or  thing  that  is  divided 
into  equal  parts. 

A  Fractional  Unit  is  one  of  the  equal  parts  into  which  the  num- 
ber or  thing  is  divided. 

The  Terms  of  a  Fraction  are  its  numerator  and  its  denominator. 

The  Denominator  of  a  fraction  shows  into  how  many  parts  the 
unit  i-s  divided. 

The  Numerator  of  a  fraction  shows  how  many  of  the  parts  are 
taken. 

A  fraction  indicates  division  ;  the  numerator  being  the  divi- 
dend and  the  denominator  the  divisor. 

The  Value  of  a  Fraction  is  the  quotient  of  the  numerator  divided 
by  the  denominator. 

Fractions  are  divided  into  two  classes  —  Common  and  Decimal, 

A  Common  Fraction  is  one  in  which  the  unit  is  divided  into  any 
number  of  equal  parts. 

A  common  fraction  is  expressed  by  writing  the  numerator  above 
the  denominator  with  a  dividing  line  between. 


SUPPLEMENT 

Common  fractions  consist  of  three  principal  classes  —  Simple, 
Compound,  and  Complex, 

A  Simple  Traction  is  one  whose  terms  are  whole  numbers. 

A  Proper  Fraction  is  a  simple  fraction  whose  numerator  is  less 
than  its  denominator. 

An  Improper  Fraction  is  a  simple  fraction  whose  numerator 
equals  or  exceeds  its  denominator. 

A  Compound  Fraction  is  a  fraction  of  a  fraction. 

A  Complex  Fraction  is  one  having  a  fraction  in  its  numerator,  or 
in  its  denominator,  or  in  both. 

A  Mixed  Number  is  a  whole  number  and  a  fraction  written 
together. 

The  Eeciprocal  of  a  Number  is  one  divided  by  that  number. 

The  Reciprocal  of  a  Fraction  is  one  divided  by  the  fraction,  or 
the  fraction  inverted. 

PRINCIPLES.  —  Multiplying  the  numerator  or  dividing  the  de- 
nominator multiplies  the  fraction. 

Dividing  the  numerator  or  multiplying  the  denominator  divides 
the  fraction. 

Multiplying  or  dividing  both  terms  of  a  fraction  by  the  same 
number  does  not  alter  the  value  of  the  fraction. 

Reduction  of  fractions  is  changing  their  terms  without  altering 
their  value. 

To  reduce  a  Fraction  to  Higher  Terms, 

RULE.  —  Multiply  both  numerator  and  denominator  by  the  same 
number. 

To  reduce  a  Fraction  to  its  Lowest  Terms, 

RULE.  —  Divide  both  terms  of  the  fraction  by  their  greatest 
common  divisor. 

A  fraction  is  in  its  lowest  terms  when  the  numerator  and  the 
denominator  are  prime  to  each  other. 


DEFINITIONS,    PRINCIPLES, 

To  reduce  a  Mixed  Number  to  an  Improper  Traction, 
RULE.  —  Multiply  the  whole  number  by  the  denominator ;  to  the 
product  add  the  numerator ;  and  place  the  sum  over  the  denom- 
inator. 

To  reduce  an  Improper  Traction  to  a  Whole  or  to  a  Mixed  Number, 
RULE.  —  Divide  the  numerator  by  the  denominator. 
A  Common  Denominator  is  a  denominator  common  to  two  or 
more  fractions. 

The  Least  Common  Denominator  is  the  smallest  denominator 
'common  to  two  or  more  fractions. 

To  reduce  Tractions  to  their  Least  Common  Denominator. 

RULE.  —  Find  the  least  common  multiple  of  all  the  denomi- 
nators for  the  least  common  denominator.  Divide  this  multiple  by 
the  denominator  of  each  fraction,  and  multiply  the  numerator  by 
the  quotient. 

ADDITION 'OF  FRACTIONS 

PRINCIPLE.  —  Only  like  fractions  can  be  added. 

RULE.  — Reduce  the  fractions,  if  necessary,  to  a  common  denom- 
inator, and  over  it  write  the  sum  of  the  numerators. 

If  there  are  mixed  numbers,  add  the  fractions  and  the  whole 
numbers  separately,  and  unite  the  results. 

SUBTRACTION  OF  FRACTIONS 

PRINCIPLE.  —  Only  like  fractions  can  be  subtracted. 

RULE.  —  Reduce  the  fractions,  if  necessary,  to  a  common  denom- 
inator, and  over  it  write  the  difference  between  the  numerators. 

If  there  are  mixed  numbers  subtract  the  fractions  and  the  whole 
numbers  separately,  and  unite  the  results. 

MULTIPLICATION   OF   FRACTIONS 

RULE.  —  Reduce  whole  and  mixed  numbers  to  improper  frac- 
tions ;  cancel  the  factors  common  to  numerators  and  denomina- 
tors, and  write  the  product  of  the  remaining  factors  in  the  numer- 
ators over  the  product  of  the  remaining  factors  in  the  denominators. 


SUPPLEMENT 


DIVISION    OF  FRACTIONS 

RULES.  —  I.  Reduce  whole  and  mixed  numbers  to  improper 
fractions.  Reduce  the  fractions  to  a  common  denominator.  Divide 
the  numerator  of  the  dividend  by  the  numerator  of  the  divisor. 

II.  Invert  the  divisor  and  proceed  as  in  multiplication  of  frac- 
tions. 

To  reduce  a  Complex  Fraction  to  a  Simple  One. 

RULES.  —  I.  Multiply  the  numerator  of  the  complex  fraction 
by  its  denominator  inverted. 

II.  Multiply  both  terms  by  the  least  common  multiple  of  the 
denominators. 

DECIMALS 

A  Decimal  Fraction  is  one  in  which  the  unit  is  divided  into 
tenths,  luindredths,  thousandths,  etc. 

A  Decimal  is  a  decimal  traction  whose  denomination  is  indi- 
cated by  the  number  of  places  at  the  right  of  the  decimal  point. 

The  Decimal  Point  is  the  mark  used  to  locate  units. 

A  Mixed  Decimal  is  a  whole  number  and  a  decimal  written 
together. 

A  Complex  Decimal  is  a  decimal  with  a  common  fraction 
written  at  its  right. 

To  write  Decimals. 

RULE.  —  Write  the  numerator  ;  and  from  the  right,  point  off  as 
many  decimal  places  as  there  are  ciphers  in  the  denominator, 
prefixing  ciphers,  if  necessary,  io  make  the  required  number. 

To  read  Decimals. 

RULE. — Read  the  numerator,  and  give  the  name  of  the  right- 
hand  order. 

PRINCIPLES.  —  Prefixing  ciphers  to  a  decimal  diminishes  its 
value. 


DEFINITIONS,    PRINCIPLES,    AND    RULES  XL 

Removing  ciphers  from  the  left  of  a  decimal  increases  its  value. 
Annexing  ciphers  to  a  decimal  or  removing  ciphers  from  its 
right  does  not  alter  its  value. 

To  reduce  a  Decimal  to  a  Common  Fraction. 

RULE. —  Write  the  figures  of  the  decimal  for  the  numerator,  and 
1,  with  as  many  ciphers  as  there  are  places  in  the  decimal,  for  the 
denominator,  and  reduce  the  fraction  to  its  lowest  terms. 

To  reduce  a  Common  Fraction  to  a  Decimal, 

RULE.  —  Annex  decimal  ciphers  to  the  numerator,  and  divide  it 
by  the  denominator. 

To  reduce  Decimals  to  a  Common  Denominator, 

RULE.  —  Make  their  decimal  places  equal  by  annexing  ciphers. 

ADDITION  AND  SUBTRACTION  OF  DECIMALS 
Decimals  are  added  and  subtracted  the  same  as  whole  numbers. 

MULTIPLICATION  OF  DECIMALS 

RULE. — Multiply  as  in  whole  numbers,  and  from  the  right  of 
the  product,  point  off  as  many  decimal  places  as  there  are  decimal 
places  in  both  factors. 

DIVISION  OF  DECIMALS 

RULE.  —  Make  the  divisor  a  whole  number  by  removing  the 
decimal  point,  and  make  a  corresponding  change  in  the  dividend. 
Divide  as  in  whole  numbers,  and  place  the  decimal  point  in  the 
quotient  under  (or  over)  the  new  decimal  point  in  the  dividend. 

ACCOUNTS  AND  BILLS 

A  Debtor  is  a  person  who  owes  another. 

A  Creditor  is  a  person  to  whom  a  debt  is  due. 


Xii  SUPPLEMENT 

An  Account  is  a  record  of  debits  and  credits  between  persons 
doing  business. 

The  Balance  of  an  account  is  the  difference  between  the  debit 
and  credit  sides. 

A  Bill  is  a  written  statement  of  an  account. 

An  Invoice  is  a  written  statement  of  items,  sent  with  merchan- 
dise. 

A  Keceipt  is  a  written  acknowledgment  of  the  payment  of 
part  or  all  of  a  debt. 

A  bill  is  receipted  when  the  words,  "  Received  Payment,"  are 
written  at  the  bottom,  signed  by  the  creditor,  or  by  some  person 
duly  authorized. 

DENOMINATE  NUMBERS 

A  Measure  is  a  standard  established  by  law  or  custom,  by 
which  distance,  capacity,  surface,  time,  or  weight  is  determined. 

A  Denominate  Unit  is  a  unit  of  measure. 

A  •  Denominate  Number  is  a  denominate  unit  or  a  collection  of 
denominate  units. 

A  Simple  Denominate  Number  consists  of  denominate  units  of 
one  kind. 

A  Compound  Denominate  Number  consists  of  denominate  units  of 
two  or  more  kinds. 

A  Denominate  Fraction  is  a  fraction  of  a  denominate  number. 

A  denominate  fraction  may  be  either  common  or  decimal, 

Eeduction  of  denominate  numbers  is  changing  them  from  one 
denomination  to  another  without  altering  their  value. 

Eeduction  Descending  is  changing  a  denominate  number  to  one 
of  a  lower  denomination. 

RULE.  —  Multiply  the  highest  denomination  by  the  number  re- 
quired to  reduce  it  to  the  next  lower  denomination,  and  to  the  prod- 
uct add  the  units  of  that  lower  denomination,  if  any.  Proceed 
in  this  manner  until  the  required  denomination  is  reached. 


DEFINITIONS,    PRINCIPLES,    AND   RULES  xiii 

Keduction  Ascending  is  changing  a  denominate  number  to  one  of 
a  higher  denomination. 

RULE.  —  Divide  the  given  denomination  successively  by  the 
numbers  that  will  reduce  it  to  the  required  denomination.  To  this 
quotient  annex  the  several  remainders. 

To  find  the  Time  between  Dates, 

RULE.  —  When  the  time  is  less  than  one  year,  find  the  exact 
number  of  days ;  if  greater  than  one  year,  find  the  time  by  com' 
pound  subtraction,  taking  30  days  to  the  month. 

PERCENTAGE 

Per  Cent  means  hundredths. 

Percentage  is  computing  by  hundredths. 

The  elements  involved  in  percentage  are  the  Base,  Bate,  Per- 
centage, Amount,  and  Difference. 

The  Base  is  the  number  of  which  a  number  of  hundredths  is 
taken. 

The  Eate  indicates  the  number  of  hundredths  to  be  taken. 

The  Percentage  is  one  or  more  hundredths  of  the  base. 

The  Amount  is  the  base  increased  by 'the  percentage. 

The  Difference  is  the  base  diminished  by  the  percentage. 

To  find  the  Percentage  when  the  Base  and  Eate  are  Given, 
RULE.  —  Multiply  the  base  by  the  rate  expressed  as  hundredths. 
To  find  the  Eate  when  the  Percentage  and  Base  are  Given, 
RULE.  —  Divide  the  percentage  by  the  base. 
To  find  the  Base  when  the  Percentage  and  Eate  are  Given, 
RULE.  —  Divide  the  percentage  by  the.  rate  expressed  as  hun- 
dredths. 

To  find  the  Base  when  the  Amount  and  Eate  are  Given. 
RULE.  —  Divide  the  amount  by  1  +  the  rate  expressed  as  hun- 
dredths. 


XIV  SUPPLEMENT 

To  find  the  Base  when  the  Difference  and  Rate  are  Given, 
RULE.  —  Divide  the  difference  by  I  — the  rate  expressed  as  hun- 
dredths. 

PROFIT  AND  LOSS 

Profit  or  Loss  is  the  difference  between  the  buying  and  selling 
prices. 

In  Profit  and  Loss, 

The  buying  price,  or  cost,  is  the  base. 
The  rate  per  cent  profit  or  loss  is  the  rate. 
The  profit  or  loss  is  the  percentage. 

The  selling  price  is  the  amount  or  difference,  according  as  it 
is  more  or  less  than  the  buying  price. 

COMMERCIAL  DISCOUNT 

Commercial  Discount  is  a  percentage  deducted  from  the  list 
price  of  goods,  the  face  of  a  bill,  etc. 

The  Met  Price  of  goods  is  the  sum  received  for  them. 

In  Commercial  Discount, 

The  list  price,  or       "|  . 

The  face  of  the  bill  j  1S  the  base' 

The  rate  per  cent  discount  is  the  rate. 

The  discount  is  the  percentage. 

The  list  price  diminished  by  the  discount  is  the  difference. 

In  successive  discounts,  the  first  discount  is  made  from  the  list 
price  or  the  face  of  the  bill ;  the  second  discount,  from  the  list 
price  or  face  of  the  bill  diminished  by  the  first  discount ;  and  so 
on. 

COMMISSION 

Commission  is  a  percentage  allowed  an  agent  for  his  services. 
A  Commission  Agent  is  one  who  transacts  business  on   com- 
mission. 


>  ig 
) 


DEFINITIONS,   PRINCIPLES,    AND   RULES  XV 

A  Consignment  is  the  merchandise  forwarded  to  a  commission 
agent. 

The  Consignor  is  the  person  who  sends  the  merchandise. 

The  Consignee  is  the  person  to  whom  the  merchandise  is  sent. 

The  Net  Proceeds  is  the  sum  remaining  after  all  charges  have 
been  deducted. 

In  buying,  the  commission  is  a  percentage  of  the  buying  price; 
in  selling,  a  percentage  of  the  selling  price;  in  collecting,  a  per- 
centage of  the  sum  collected;  hence: 

The  sam  invested,  or 

The  sum  collected 

The  rate  per  cent  commission  is  the  rate. 

The  commission  is  the  percentage. 

The  sum  -invested  increased  by  the  commission  is  the  amount. 

The  sum  collected  diminished  by  the  commission  is  the  differ- 
ence. 

INSURANCE 

Insurance  is  a  contract  of  indemnity. 

Insurance  is  of  three  kinds  —  Tire,  Marine,  and  Life, 

Fire  Insurance  is  indemnity  against  loss  of  property  by  fire. 

Marine  Insurance  is  indemnity  against  loss  of  property  by  the 
casualities  of  navigation. 

Life  Insurance  is  indemnity  against  loss  of  life. 

The  Insurance  Policy  is  the  contract  setting  forth  the  liability  of 
the  insurer. 

The  Policy  Pace  is  the  amount  of  insurance. 

The  Premium  is  the  price  paid  for  insurance. 

The  Insurer,  or  Underwriter,  is  the  company  issuing  the  policy. 

The  Insured  is  the  person  for  whose  benefit  the  policy  is  issued. 

In  Insurance, 

The  policy  face  is  the  base. 

The  rate  per  cent  premium  is  the  rate. 

The  premium  is  the  percentaae. 


XVl  SUPPLEMENT 

TAXES 

A  Tax  is  a  sum  of  money  levied  on  persons  or  property  foi 
public  purposes. 

A  Personal,  or  Poll  Tax,  is  a  tax  on  the  person. 

A  Property  Tax  is  a  tax  of  a  certain  per  cent  on  the  assessed 
value  of  property. 

Property  may  be  either  personal  or  real. 

Personal  Property  consists  of  such  things  as  are  movable. 

Eeal  Property  is  that  which  is  fixed,  or  immovable. 

In  Taxes, 

The  assessed  value  is  the  base. 
The  rate  of  taxation  is  the  rate. 
The  tax  is  the  percentage, 

DUTIES 

Duties  are  taxes  on  imported  goods. 
Duties  are  either  Specific  or  Ad  Valorem. 
A  Specific  Duty  is  a  tax  on  goods  without  regard  to  cost. 
An  Ad  Valorem  duty  is  a  tax  of  a  certain  per  cent  on  the  cost 
of  goods. 

In  Ad  Valorem  Duties, 
The  cost  of  the  goods  is  the  base. 
The  rate  per  cent  duty  is  the  rate. 
The  ad  valorem  duty  is  the  percentage. 

INTEREST 

Interest  is  the  sum  paid  for  the  use  of  money. 

The  Principal  is  the  sum  loaned. 

The  Amount  is  the  sum  of  the  principal  and  interest. 

The  Bate  of  Interest  is  the  rate  per  cent  for  one  year. 

The  Legal  Bate  is  the  rate  fixed  by  law. 

"Usury  is  interest  at  a  higher  rate  than  that  fixed  by  law. 

Simple  Interest  is  interest  on  the  principal  only. 


DEFINITIONS,    PRINCIPLES,    AND   RULES 

To  find  the  Interest  when  the  Principal,  Time,  and  Eate  are  Given. 

RULE.  —  Multiply  the  principal  by  the  rate  expressed  as  hun- 
dredths,  and  this  product  by  the  time  expressed  in  years. 

To  find  the  Time  when  the  Principal,  Interest,  and  Kate  are  Given. 
RULE.  —  Divide  the  given  interest  by  the  interest  for  one  year. 
To  find  the  Eate  when  the  Principal,  Interest,  and  Time  are  Given. 

RULE.  —  Divide  the  given  interest  by  the  interest  at  one  per 
cent. 

To  find  the  Principal  when  the  Interest,  Eate,  and  Time  are  Given, 

RULE.  —  Divide  the  given  interest  by  the  interest  on  $  1. 

To  find  the  Principal  when  the  Amount  and  Time  and  Eate  are 
Given, 

RULE.  —  Divide  the  given  amount  by  the  amount  of  $1. 

LNTEKEST  BY  ALIQUOT  PARTS. 

To  find  the  Interest  for  Tears,  Months,  and  Days. 

RULE.  —  Find  the  interest  for  one  year  and  take  this  as  many 
times  as  there  are  years. 

Take  the  greatest  number  of  the  given  months  that  equals  an 
aliquot  part  of  a  year  and  find  the  interest  for  this  time.  Take 
aliquot  parts  of  this  for  the  remaining  months. 

In  the  same  manner  find  the  interest  for  the  days. 

The  sum  of  these  interests  will  be  the  interest  required. 

To  find  the  Intarest  when  the  Time  is  Less  than  a  Tear, 

RULE.  —  Find  the  interest  for  the  time  in  months  or  days  that 
will  gain  one  per  cent  of  the  principal. 

Find  by  aliquot  parts,  as  in  the  first  rule,  the  interest  for  the 
remaining  time. 

The  sum  of  these  interests  will  be  the  interest  required. 


XViii  SUPPLEMENT 

INTEREST  BY  Six  PER  CENT  METHOD. 

To  find  the  Interest  at  6%. 

RULE.  —  For  Years :  Multiply  the  principal  by  the  rate  ex- 
pressed as  hundredth^,  and  that  product  by  the  number  of  years. 

For  Months :  Move  the  decimal  point  two  places  to  the  left,  and 
'multiply  by  one- half  the  number  of  months. 

For  Days :  Move  the  decimal  point  three  places  to  the  left,  and 
multiply  by  one-sixth  the  number  of  days. 

To  find  the  interest  at  any  other  rate  per  cent,  divide  the  in- 
terest at  6%  by  6,  and  multiply  the  quotient  by  the  given  rate. 

To  find  Exact  Interest. 

RULE.  —  Multiply  the  principal  by  the  rate  expressed  as  hun- 
dredths,  and  that  product  by  the  time  expressed  in  years  of  365 
days. 

ANNUAL   INTEREST 

Annnal  Interest  is, interest  payable  annually.  If  not  paid  when 
due,  annual  interest  draws  simple  interest. 

To  find  the  Amount  Due  on  a  Note  with  Annual  Interest,  when  the 
Interest  has  not  been  Paid  Annually, 

RULE.  —  Fmd  the  interest  on  the  principal  for  the  entire  time, 
and  on  each  annual  interest  for  the  time  it  remained  unpaid. 
The  sum  of  the  principal  and  all  the  intei'est  is  the  amount  due. 

COMPOUND  INTEREST 

Compound  Interest  is  interest  on  the  principal  and  on  the  un- 
paid interest,  which  is  added  to  the  principal  at  regular  inter- 
vals. The  interest  may  be  compounded  annually,  semi-annually, 
quarterly,  etc.,  according  to  agreement. 

To  find  Compound  Interest. 

RULE. —  Ftnd  fhe  amount  of  the  given  principal  for  the  first 
period.  Considering  this  as  a  new  principal,  find  the  amount  of 


DEFINITIONS,    PRINCIPLES,   AND    RULES  xix 

it  foi  the  next  period,  continuing  in  this  manner  for  the  given 
time. 

Find  the  difference  between  the  last  amount  and  the  given 
principal,  which  will  be  the  compound  interest. 

PARTIAL  PAYMENTS 

Partial  Payments  are  part  payments  of  a  note  or  debt.  Each 
payment  is  recorded  on  the  back  of  the  note  or  the.  written 
obligation. 

UNITED  STATES  RULE.  —  Find  the  amount  of  the  principal  to 
the  time  when  the  payment  or  the  sum  of  two  or  more  payments 
equals  or  exceeds  the  interest. 

From  this  amount  deduct  the  payment  or  sum  of  payments. 

Use  the  balance  then  due  as  a  new  principal,  and  proceed  as 
before. 

MERCHANTS'  RULE.  —  Find  the  amount  of  an  interest-bearing 
note  at  the  time  of  settlement. 

Find  the  amount  of  each  credit  from  its  time  of  payment  to  the 
time  of  settlement ;  subtract  their  sum  from  the  amount  of  the 
principal. 

BANK  DISCOUNT 

Bank  Discount  is  a  percentage  retained  by  a  bank  for  advanc- 
ing money  on  a  note  before  it  is  due. 

The  Sum  Discounted  is  the  face  of  the  note,  or  if  interest-bear- 
ing, the  amount  of  the  note  at  maturity. 

The  Term  of  Discount  is  the  number  of  days  from  the  day  of 
discount  to  the  day  of  maturity. 

The  Bank  Discount  is  the  interest  on  the  sum  discounted  for 
the  term  of  discount. 

The  Proceeds  of  a  note  is  the  sum.  discounted  less  the  bank  dis- 
count. 

Problems  in  bank  discount  are  calculated  as  problems  in 
interest. 


XX  SUPPLEMENT 

In  Bank  Discount, 

The  sum  discounted  is  the  principal. 

The  rate  of  discount  is  the  rate  of  interest. 

The  term  of  discount  is  the  time. 

The  bank  discount  is  the  proceeds. 

EXCHANGE 

Exchange  is  making  payments  at  a  distance  by  means  of  drafts 
or  bills  of  exchange. 

Domestic  Exchange  is  exchange  between  places  in  the  same 
country. 

Foreign  Exchange  is  exchange  between  different  countries. 

Exchange  is  at  par  when  a  draft,  or  bill,  sells  for  its  face 
value ;  at  a  premium  when  it  sells  for  more  than  its  face  value ; 
at  a  discount  when  it  sells  for  less. 

The  cost  of  a  sight  draft  is  the  face  of  the  draft  increased  by 
the  premium,  or  diminished  by  the  discount. 

The  cost  of  a  time  draft  is  the  face  of  the  draft  increased  by 
the  premium,  or  diminished  by  the  discount,  and  this  result, 
diminished  by  the  bank  discount. 

To  find  the  Cost  of  a  Draft 

KULE.  —  Find  the  cost  of  $  1  of  the  draft;  multiply  this  ly  the 
face  of  the  draft. 

To  find  the  Face  of  a  Draft. 

RULE.  —  Divide  the  cost  of  the  draft  by  the  cost  of  $1  of  the 
draft. 

EQUATION  OF  PAYMENTS 

Equation  of  Payments  is  a  method  of  ascertaining  at  what  time 
several  debts  due  at  different  times  may  be  settled  by  a  single 
payment. 

The  Equated  Time  of  payment  is  the  time  when  the  several 
debts  may  be  equitably  settled  by  one  payment. 

The  Term  of  Credit  is  the  time  the  debt  has  to  run  before  it 
becomes  due. 


DEFINITIONS,    PRINCIPLES,   AND   RULES  XXi 

The  Average  Term  of  Credit  is  the  time  the  debts  due  at  different 
times  have  to  run,  before  they  may  be  equitably  settled  by  one 
payment. 

To  find  the  Equated  Time  of  Payment  when  the  Terms  of  Credit 
begin  at  the  Same  Date, 

RULE.  —  Multiply  each  debt  by  its  term  of  credit,  and  divide  the 
sum  of  the  products  by  the  sum  of  the  debts.  The  quotient  will  be 
the  average  term  of  credit. 

Add  the  average  term  of  credit  to  the  date  of  the  debts,  and  the 
result  will  be  the  equated  time  of  payment. 

To  find  the  Equated  Time  when  the  Terms  of  Credit  begin  at 
Different  Dates, 

RULE.  — Find  the  date  at  which  each  debt  becomes  due.  Select 
the  earliest  date  as  a  standard. 

Multiply  each  debt  by  the  number  of  days  between  the  standard 
date  and  the  date  when  the  debt  becomes  due,  and  divide  the  sum 
of  the  products  by  the  sum  of  the  debts.  The  quotient  will  be  the 
average  term  of  credit  from  the  standard  date. 

Add  the  average  term  of  credit  to  the  standard  date,  and  the 
result  will  be  the  equated  time  of  payment. 

RATIO 

Ratio  is  the  relation  one  number  bears  to  another  of  the  same 
kind. 

The  Terms  of  the  ratio  are  the  numbers  compared. 

The  Antecedent  is  the  first  term. 

The  Consequent  is  the  second  term. 

The  antecedent  and  consequent  form  a  couplet. 

PRINCIPLES.  —  See  Fractions. 

PROPORTION 

A  Proportion  is  formed  by  two  equal  ratios. 

The  Extremes  of  a  proportion  are  the  first  and  last  terms. 

The  Means  of  a  proportion  are  the  second  and  third  terms. 


XX11  SUPPLEMENT 

PRINCIPLES.  —  The  product  of  the  means  is  equal  to  the  prod- 
uct of  the  extremes. 

Either  mean  equals  the  product  of  the  extremes  divided  by  the 
other  mean. 

Either  extreme  equals  the  product  of  the  means  divided  by  the 
other  extreme. 

RULE  FOR  PROPORTION.  —  Represent  the  required  term  by  x. 

Arrange  the  terms  so  that  the  required  term  and  the  similar 
known  term  may  form  one  couplet,  the  remaining  terms  the  other. 

If  the  required  term  is  in  the  extremes,  divide  the  product  of  the 
means  by  the  given  extreme. 

If  the  required  term  is  in  the  means,  divide  the  product  of  the 
extremes  by  the  given  mean. 

PARTNERSHIP 

Partnership  is  an  association  of  two  or  more  persons  for  busi- 
ness purposes. 

The  Partners  are  the  persons  associated. 

The  Capital  is  that  which  is  invested  in  the  business. 

The  Assets  are  the  partnership  property. 

The  Liabilities  are  the  partnership  debts. 

To  find  the  Profit,  or  Loss,  of  Each  Partner  when  the  Capital  of 
Each  is  Employed  for  the  Same  Period  of  Time, 

RULE.  —  Find  the  part  of  the  entire  profit,  or  loss,  that  each 
partner's  capital  is  of  the  entire  capital. 

To  find  the  Profit,  or  Loss,  of  Each  Partner  when  the  Capital  of 
Each  is  Employed  for  Different  Periods  of  Time, 

RULE. — Find  each  partner's  capital  for  one  month,  by  multi- 
plying the  amount  he  invests  by  the  number  of  months  it  is 
employed;  then  find  the  part  of  the  entire  profit,  or  loss,  that  each 
partner's  capital  for  one  month  is  of  the  entire  capital  for  one 
month. 


DEFINITIONS,    PRINCIPLES,   AND   RULES 


INVOLUTION 

A  Power  of  a  number  is  the  product  obtained  by  using  that 
number  a  certain  number  of  times  as  a  factor. 

The  First  Power  of  a  number  is  the  number  itself. 

The  Second  Power  of  a  number,  or  the  Square,  is  the  product  of 
a  number  taken  twice  as  a  factor. 

The  Third  Power  of  a  number,  or  the  Cubs,  is  the  product  of  a 
number  taken  three  times  as  a  factor. 

An  Exponent  is  a  small  figure  written  a  little  to  the  right  of  the 
upper  part  of  a  number  to  indicate  the  power. 

Involution  is  finding  any  power  of  a  number. 

To  find  the  Power  of  a  Number, 

RULE.  —  Take  the  number  as  a  factor,  as  many  times  as  there 
are  units  in  the  exponent. 

EVOLUTION 

A  Eoot  is  one  of  the  equal  factors  of  a  number. 

The  Square  Eoot  of  a  number  is  one  of  its  two  equal  factors. 

The  Cube  Eoot  of  a  number  is  one  of  its  three  equal  factors. 

Evolution  is  finding  any  root  of  a  number. 

Evolution  may  be  indicated  in  two  ways  :  by  the  Radical 
Sign,  V",  or  by  a  fractional  exponent. 

The  Index  of  a  root  is  a  small  figure  placed  a  little  to  the  left 
of  the  upper  part  of  the  radical  sign,  to  indicate  what  root  is  to 
be  found.  In  expressing  square  root,  the  index  is  omitted. 

In  the  fractional  exponent,  the  numerator  indicates  the  power 
to  which  the  number  is  to  be  raised;  the  denominator  indicates 
the  root  to  be  taken  of  the  number  thus  raised. 

To  find  the  Square  Eoot  of  a  Number, 

RULE.  —  Point  off  in  periods  of  two  figures,  commencing  at 
units.  Find  the  greatest  square  in  the  first  period  and  place  the 
root  in  the  quotient.  Subtract  this  square  from  the  first  period, 
and  bring  down  the  next  period. 


XXIV  SUPPLEMENT 

Multiply  the  quotient  figure  by  two,  and  use  it  as  a  trial  divisor. 
Place  the  second  figure  in  the  quotient,  and  annex  it  also  to  the 
trial  divisor.  Then  multiply  the  figures  in  the  trial  divisor  by  the 
second  quotient  figure,  and  subtract. 

Bring  down  the  next  period,  and  proceed  as  before  until  the 
square  root  is  found. 

To  find  the  Square  Boot  of  a  Traction. 

RULE.  —  Reduce  the  fraction  to  its  simplest  form,  and  find  the 
square  root  of  each  term  separately. 

To  find  the  Onbe  Eoot  of  a  Number. 

RULE.  —  Point  off  in  periods  of  three  figures  each,  beginning  at 
units. 

Find  the  greatest  cube  in  the  first  period  and  place  the  root  in 
the  quotient.  Subtract  this  cube  from  the  first  period,  and  bring 
down  the  next  period. 

Multiply  the  square"  of  the  first  quotient  figure  by  three  and 
annex  two  ciphers  for  a  trial  divisor.  Place  the  second  figure  in 
the  quotient.  Then,  to  the  trial  divisor  add  three  times  the  prod- 
uct of  the  first  and  second  figures,  also  the  square  of  the  second. 
Multiply  this  su?n  by  the  second  figure  and  subtract. 

Bring  down  the  next  period,  and  proceed  as  before  until  the  cube 
root  is  found. 

To  find  the  Cube  Eoot  of  a  Traction. 

RULE.  —  Reduce  the  fraction  to  its  simplest  form,  and  find  the 
cube  root  of  each  term  separately. 

STOCKS  AND  BONDS. 

Capital  Stock  is  the  money  or  property  employed  by  a  corpora- 
tion in  its  business. 

A  Share  is  one  of  the  equal  divisions  of  capital  stock. 

The  Stockholders  are  the  owners  of  the  capital  stock. 

The  Par  Value  of  stock  is  the  face  value. 

The  Market  Value  of  stock  is  the  sum  for  which  it  may  be  sold. 


DEFINITIONS,   PRINCIPLES,   AND   RULES  XXV 

Stock  is  at  a  premium  when  the  market  value  is  above  the 
par  value ;  at  a  discount,  when  below  par. 

Bonds  are  interest-bearing  notes  issued  by  a  government  or  a 
corporation. 

A  Dividend  is  a  percentage  apportioned  among  the  stockholders. 
A  Stock  Broker  is  a  person  who  deals  in  stocks. 
Brokerage  is  a  percentage  allowed  a  stock  broker  for  his  services. 
In  Stocks  and  Bonds, 

The  par  value  is  the  base. 

The  rate  per  cent  premium,  or  discount,  is  the  rate. 

The  premium,      ~) 

discount,  or   >  is  the  percentage. 
dividend       J 

mi  i    j      i      •    M     f  amount,  or 

The  market  value  is  the  «    7 .  ~ 

(  difference. 


NOTES,  DRAFTS,  AND  CHECKS. 

A  Promissory  Note  is  a  written  promise  to  pay  a  specified  sum 
on  demand,  or  at  a  specified  time. 

The  Pace  of  a  note  is  the  sum  named  in  the  note. 

The  Maker  is  the  person  who  signs  it. 

The  Payee  is  the  person  to  whom  the  sum  specified  i$  to  be 
paid. 

The  Indorser  is  the  person  who  signs  his  name  on  the  back  of 
the  note,  thus  becoming  liable  for  its  payment  in  case  of  default 
of  the  maker. 

An  Interest-bearing  Note  is  one  payable  with  interest. 

If  the  words  "  with  interest "  are  omitted,  interest  cannot  be 
collected  until  after  maturity. 

A  Demand  Note  is  one  payable  when  demand  of  payment  is 
made. 

A  Time  Note  is  one  payable  at  a  specified  time. 

A  Joint  Note  is  one  signed  by  two  or  more  persons  who  jointly 
promise  to  pay. 


XXVI  SUPPLEMENT 

A  Joint  and  Several  Note  is  one  signed  by  two  or  more  persons 
who  jointly  and  severally  promise  to  pay. 

In  a  joint  note,  each  person  is  liable  for  the  whole  amount, 
but  they  must  all  be  sued  together.  In  the  joint  and  several 
note,  each  is  liable  for  the  whole  amount,  and  may  be  sued 
separately. 

A  Negotiable  Note  is  one  that  may  be  transferred  or  sold.  It 
contains  the  words  "  or  bearer,"  or  "  or  order." 

A  Non-negotiable  Note  is  one  not  payable  to  the  bearer,  nor  to 
the  payee's  order. 

The  Matnrity  of  a  note  is  the  day  on  which  it  legally  falls  due. 

A  Draft,  or  Bill  of  Exchange,  is  a  written  order  directing  the 
payment  of  a  specified  sum  of  money. 

The  Face  of  a  draft  is  the  sum  named  in  it. 

The  Drawer  is  the  person  who  signs  the  draft. 

The  Drawee  is  the  person  ordered  to  pay  the  sum  specified. 

The  Payee  is  the  person  to  whom  the  sum  specified  is  to  be 
paid. 

A  Sight  Draft  is  one  payable  when  presented. 

A  Time  Draft  is  one  payable  at  a  specified  time. 

An  Acceptance  of  a  time  draft  is  an  agreement  by  the  drawee 
to  pay  the  draft  at  maturity,  which  he  signifies  by  writing  across 
the  face  of  the  draft  the  word  "  accepted  "  with  the  date  and  his 
name. 

A  Check  is  an  order  on  a  bank  or  banker  to  pay  a  specified 
sum  of  money. 


PRIMAEY   ARITHMETIC 
ANSWERS. 


Page  72. 

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985. 

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6948. 

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40. 

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1008. 

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Page  73. 

32. 

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11. 

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106.  8. 

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48.  1923. 

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107.  7. 

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114.  155T\V 

108.  11. 

147.  24f. 

Page  116. 

115.  1036&. 

109.  124. 

148.  52£. 

58.  36,&. 

116.  1014ff. 

110.  124. 

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111.  243. 

Page  115. 

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112.  129. 

1.  450. 

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Page  117. 

113.  420. 

2.  640. 

62.  126ft. 

1.  34J. 

8 


ANSWERS. 


2.  48$. 

2.  299. 

43.  6656. 

84.  91,696. 

3.  75$. 

3.  793. 

44.  6630. 

85.  93,005. 

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4.  715. 

45.  8208. 

86.  99,648. 

6.  83. 

5.  196. 

46.  9782. 

87.  99,425. 

6.  42J. 

6.  322. 

47.  9620. 

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7.  62f. 

7.  854. 

48.  9375. 

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54.  10,323. 

95.  72,415. 

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20.  96. 

20.  936. 

61.  15,167. 

Page  120. 

21.  3$. 

21.  756. 

62.  22,104. 

1.  13. 

22.  17$. 

22.  840. 

63.  11,433. 

2.  21. 

23.  5. 

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64.  15,580. 

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25.  968. 

66.  28,980. 

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26.  52$. 

26.  736. 

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6.  211. 

27.  7$. 

27.  576. 

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7.  123. 

28.  58|. 

28.  858. 

69.  22,050. 

8.  222. 

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29.  1024. 

70.  31,360. 

9.  11. 

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71.  5814. 

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31.  1496. 

72.  11,948. 

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32.  1575. 

73.  18,408. 

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33.  1806. 

74.  27,456. 

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78.  69,160. 

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Page  119. 

41.  4464. 

82.  90,300. 

21.  11. 

1.  182. 

42.  5544. 

83.  95,961. 

22.  12. 

ANSWERS. 


9 


23.    13. 

4.    66f. 

Page  128. 

19.    60  miles. 

24.   23. 

5.   81f. 

51.    $916.61. 

20.    150  days. 

25.   31. 

6.   98|. 

52.   $778.91. 

26.    112. 

7.   91f. 

53.   f  1780.53}. 

1.    31. 

27.   213. 

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54.    $3431.48*. 

2.    31. 

28.   313. 

9.    99f. 

55.   $76.11. 

3.   24. 

29.    211. 

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57.   $657.66. 

4.    31. 

30.    122. 

58.   $4.17. 

5.   42. 

31.    311. 

Page  127. 

59.   $76.50. 

6.    23. 

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11.    42f. 

61.   $58.20. 

7.   41. 

33.   311. 

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62.  $457.12. 

8.    11. 

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13.    78. 

63.   $977.67. 

9.    21. 

35.   33. 

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64.    $963.69. 

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11.    11. 

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68.  $108.06. 

14.   21. 

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69.   $37.12£. 

15.    31. 

41.   31. 

20.   80.     ' 

70.   $.34. 

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Page  130. 

18.   31. 

Page  122. 

27.  3£. 

1.   $4.14. 

19.    111. 

1.    200  feet. 

28.    18J. 

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20.    111. 

2.   $20.70. 

29.   35£. 

3.  $14. 

21.    321. 

3.    11  sheep. 

30.   53£. 

4.    21  cents. 

22.    322. 

4.    900  inches. 

31.    88f 

5.   $252. 

23.    300. 

5.    480  ounces. 

32.    7£. 

6.   $750. 

24.    302^. 

6.    96  cents. 

33.   37f. 

7.   5  yards. 

7.    16  pages. 

34.    15£. 

8.    96  jars. 

Page  132. 

8.    96  packages 

.     35.    21|. 

9.   9  gallons. 

25.    20&. 

9.    25  gallons. 

36.   27^. 

10.   75  cents. 

26.    40|f 

10.    7  ounces. 

37.    25f. 

11.   $1620. 

27.    50&. 

11.   $50. 

38.   36£. 

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28.    203. 

12.   $100. 

39.    8}. 

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40.    39|. 

30.    20%. 

14.    15  miles. 

41.    17f. 

Page  131. 

31.    202&. 

42.    37f. 

14.    68  days. 

32.    101. 

Page  126. 

43.    15f 

15.   36  bushels. 

33.    101^. 

1.   41|. 

44.    45£. 

16.   $2.56. 

34.    203. 

2.    62f. 

45.    47. 

17.   $240. 

35.    304. 

3.   57f 

60.   4f 

18.    260  feet. 

36.    200^. 

10 


ANSWERS. 


37.   304. 

25.   39. 

64.   15. 

4.   48,300. 

38.    430&. 

26.    If 

65.    13. 

5.    78,300. 

39.   203. 

27.    lOf. 

66.   200. 

6.   98,400. 

40.   431^. 

28.    lOf 

67.  48. 

7.   98,800. 

41.    202. 

29.    lOf 

68.    60. 

8.   91,000. 

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30.   10. 

69.   32. 

9.    72,000. 

43.    221T8T- 

31.   9f. 

70.   60. 

44.   123. 

32.    20£. 



Page  141. 

45.    325. 

33.   41f 

1.   $5. 

10.    90,000. 

46.    231^j. 

34.    18J. 

2.    150  stamps. 

11.    88,800. 

47.    101&. 

35.   26f 

3.    62  cows. 

12.    84,150. 

48.    34^. 

36.    31f. 

4.    $16. 

15.    95,000. 

49.    122£f 

37.    67f. 

5.    28  pounds. 

16.    77,400. 

50.    103. 

38.    19f 

17.   83,700. 

51.    20&. 

39.    If 

Page  139. 

18.    89,100. 

40.   19f. 

6.    144  pieces. 

19.   93,000. 

Page  137. 

7.  $12.80. 

20.    67,200. 

1.   47. 

Page  138. 

8.   40  boxes. 

21.    95,370. 

2.   84. 

41.    185. 

9.   49  inches. 

22.    99,540. 

3.   32. 

42.    48. 

10.   234  eggs. 

23.    81,480. 

4.    73. 

43.    203. 

11.   $60. 

24.    88,480. 

5.    81f 

44.    19. 

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6.    68£. 

45.    90. 

13.    15  cents. 

Page  143. 

7.    38f 

46.    29. 

14.   $2.94. 

1.  7f. 

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2.   9f 

9.    70f. 

48.   4.       . 

16.    7  packages. 

3.    16f 

10.    83f 

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17.  $3. 

4.    19f 

11.    6f 

50.    17,376. 

18.    750  pounds. 

5.    20fr. 

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51.    1000. 

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6.   19f 

13.    19|. 

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7.   48. 

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8.   37. 

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54.    78. 

9.    22}. 

16.    26. 

55.    126. 

Page  140. 

10.   45f. 

17.   31f 

56.    168. 

22.    30  cents. 

11.   84f 

18.    13f 

57.    144. 

23.    5  yards. 

12.    89f 

19.    lOf 

58.    144. 

24.    196  pounds. 

13.    llf. 

20.    22f. 

59.    144. 

25.    20  pieces. 

14.    39f. 

21.    2f 

60.    84. 



15.    48f 

22.    13f 

61.   10. 

1.    70,800. 

16.   7. 

23.    21. 

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17.   22f 

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126. 

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32. 

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75. 

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Page  145. 

41. 

304. 

82. 

206¥87. 

1. 

54. 

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2. 

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84. 

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3. 

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136. 

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545ft. 

7. 

24. 

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108. 

89. 

555ft. 

8. 

33. 

49. 

204. 

90. 

355. 

ANSWERS.  11 


91.  91. 

92.  142. 

93.  113. 

94.  112ff. 

95.  87f£. 

96.  51ff. 

97.  103$f. 

98.  102f$. 

99.  103ft. 

Page  146. 

1.  136. 

2.  32. 

3.  34. 

4.  122. 
6.  87. 

6.  75. 

7.  24. 

8.  65. 

9.  25. 
10.  49. 

Page  147. 

1.  136/&. 
2. 
3. 
4. 
5. 
6. 
7. 

8-   80^. 
9- 
10. 


12 


ANSWERS. 


28.    27|. 

16.    96,266. 

55.    92,640. 

29.    16f. 

17.   94,520. 

56.    85,181. 

30.   53f. 

18.   94,518. 

57.    82,926. 

31.   31. 

19.   90,750. 

58.   64,684. 

32.   26$. 

20.   93,396. 

59.    76,020. 

33.    21*. 

21.   96,170. 

60.   97,768. 

34.    4$. 

22.    97,908. 

61.   90,752. 

35.    44$. 

23.    89,159. 

62.    70,455. 

36.    11$. 

24.    87,472. 

63.    98,049. 

37.    8f. 

25.    97,768. 

64.   86,592. 

38.    16|. 

26.    95,918. 

65.  98,245. 

39.    24$. 

27.   43. 

66.   71,604. 

40.    32f. 

28.   78. 

67.   99,770. 

Page  149. 

41.    40f. 

29.    32. 

68.   98,802. 

1.    27f. 

42.    48f. 

30.    24. 

69.   81,804. 

2.   47$. 

43.    56$. 

31.   14. 

70.   95,081. 

3.   51f. 

44.    64f 

32.    13. 

71.   98,245. 

4.   38f. 

45.    72f 

33.    14. 

72.    98,245. 

6.   66f 

46.    79£. 

34.    12. 

73.   92,486. 

6.   99f. 

47.    27f 

35.    11. 

74.    75,072. 

7.   68£. 

48.   17f 

36.   9. 

8.    94f. 

49.    29^. 

Page  153. 

9.   95f. 

50.   62f. 

Page  152. 

75.    88. 

10.   85. 

37.   8. 

76.    105. 

11.   99. 

38.   13. 

77.   99. 

12.   23. 

Page  151. 

39.    15. 

78.   98. 

13.   44$. 

1.    99,684. 

40.    13. 

79.   69. 

14.    70f.  ' 

2.  85,731. 

41.   24. 

80.    69. 

15.   27*. 

3.   95,772. 

42.    23. 

81.    168. 

16.   42|. 

4.   94,770. 

43.   45. 

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5.   94,095. 

44.   75. 

83.   90. 

18.  98$. 

6.   89,622. 

45.    33. 

84.   93. 

19.    83f. 

7.   96,882. 

46.   22. 

85.    186. 

20.   35f. 

8.    95,914. 

47.    8. 

86.    154. 

21.    6$. 

9.   99,507. 

48.    4. 

87.   232. 

22.   23f. 

10.   91,344. 

49.   6. 

88.   368. 

23.    22$. 

11.   86,592. 

50.   4. 

89.   297. 

24.    31f. 

12.    97,020. 

51.  68,580. 

90.    100. 

25.   31$. 

13.    93,832. 

52.   96,621. 

91.    102. 

26.    12f 

14.    79,328. 

53.    96,859. 

92.   205. 

27.    10. 

15.   91,464. 

54.    96,740. 

93.   255. 

ANSWERS. 


13 


94.   320. 

33.   42,372. 

74.    155. 

9.    2£  yards. 

95.    456. 

34.    107,028. 

75.    138. 

10.   108  quarters. 

96.    675. 

35.   96,444. 

76.    123. 

11.   $72.50. 

97.    880. 

36.  92,376. 

77.    109. 

12.   3  cents. 

98.    615. 

37.    337£. 

78.   406. 

13.    86  feet. 

38.    673f. 

79.    308. 

14.    $7.80. 

Page  154. 

39.    1237£. 

80.    203. 

15.   $2.40. 

1.   360. 

40.    1897$. 

81.    170. 

16.    95  cents. 

2.    1125. 

41.    1683$. 

82.    146. 

17.   99  cents. 

3.    800. 

42.    2880f. 

83.    123. 

18.   48  eggs;  144 

4.    1200. 

43.    1869f. 

84.    105. 

eggs. 

5.   1770. 

44.   4575f. 

85.    104. 

19.    40  bushels. 

6.   2800. 

45.   6286£. 

86.    98. 

20.   $60. 

7.   2331. 

46.    21,441|. 

87.    48. 

21.    413     butter- 

8.    16,044. 

47.  40,138. 

88.    33^fo. 

flies. 

9.   14,883. 

48.   44,500f. 

89.   24. 

22.   99  cents. 

10.   39,234. 

49.   42,274f. 

90.    19. 

11.    22,243. 

50.   99,682f. 

91.   16. 

Page  160. 

12.   4400. 

51.    65,166f. 

92.   14. 

23.    130  yards. 

52.  4231. 

93.    12. 

24.   7f  acres. 

Page  155. 

53.    3152. 

94.    11. 

25.   $1.75. 

13.   6578. 

54.    2405. 

QK       QQ 

14.    23,922. 

55.    1600. 

*/«-!•       t/O. 

96.   87. 

1.   91,448. 

15.   43,190. 

56.    1623. 

97.    75. 

2.   86,400. 

16.    49,260. 

57.    1405. 

98.    33. 

3.   97,886. 

17.   17,922. 

58.    1234. 

99.   23. 

4.   90,288. 

18.   61,479. 

59.    1035. 

100.    9. 

5.    89,415. 

19.    85,200. 

60.    2305. 

101.   4. 

6.   88,971. 

20.   82,810. 

61.   2046. 

102.   8. 

7.   89,208. 

21.    6888. 

62.    1653. 

8.   82,766. 

22.    13,552. 

63.   1408. 

Page  158. 

9.   99,696. 

23.   39,528. 

64.   1305. 

1.  $7. 

10.    73,140. 

24.    51,968. 

65.    1060. 

2.   $2.82. 

11.    82,602. 

25.    14,610. 

66.   1003. 

3.    192  pints. 

12.    99,960. 

26.    25,280. 

67.   3265. 

4.   20  yards. 

13.   97,633. 

27.   50,904. 

68.    907. 

5.   $2.40. 

14.    96,348. 

28.   65,400. 

69.    807. 

6.    40  cents. 

15.    180. 

29.   84,252. 

70.   486. 

16.    232. 

30.   96,560. 

71.    325. 

Page  159. 

17.    348. 

31.   79,380. 

72.   247. 

7.    $3. 

18.   567. 

32.   30,537. 

73.   189. 

8.   $2.25. 

19.   864. 

14 


ANSWERS. 


20. 

1120. 

8. 

99*. 

6. 

538ft. 

47. 

6. 

21. 

777. 

9. 

82*. 

7. 

713*f. 

48. 

3. 

22. 

945. 

10. 

108$. 

8. 

282ff 

23. 

1100. 

11. 

64*. 

9. 

1636$f. 

Page  164. 

24. 

1343. 

12. 

91. 

10. 

1966$f 

1. 

31  barrels. 

25. 

496. 

13. 

95. 

11. 

811  A- 

2. 

6  yards. 

26. 

1454f. 

14. 

37$. 

12. 

787**. 

3. 

380  inches. 

27. 

96,000. 

15. 

26. 

13. 

478f*. 

28. 

99,712. 

14. 

222**, 

Page  165. 

29. 

96,888. 

Page 

162. 

15. 

279ff. 

4. 

H  yards. 

30. 

99,328. 

16. 

86. 

16. 

m**. 

5. 

25  cents;  $L 

31. 

77,608. 

17. 

81$. 

17. 

182*. 

6. 

16  cents. 

32. 

99,450. 

18. 

49. 

18. 

432f#f. 

7. 

98  cents. 

33. 

99,902. 

19. 

83$. 

19. 

153$ff 

8. 

3i  pounds. 

34. 

95,841. 

20. 

87$. 

20. 

86&V 

9. 

39  pints. 

35. 

61,845. 

21. 

64$. 

21. 

181^.. 

10. 

145  sheep. 

36. 

99,102. 

22. 

28$. 

22. 

113|$f. 

11. 

$2. 

37. 

96,696. 

23. 

37$. 

23. 

104f$$. 

12. 

93  cents. 

38. 

92,976. 

24. 

35*. 

24. 

HI***- 

13. 

6  weeks. 

39. 

99,051. 

25. 

54$. 

25. 

70f$f 

14. 

35  gallons. 

40. 

93,345. 

26. 

69f. 

26. 

709jfj. 

15. 

10  Ib.  5  oz. 

41. 

96,744. 

27. 

30*. 

27. 

219ft. 

16. 

$2. 

42. 

88,920. 

28. 

81$. 

28. 

132&V 

43. 

99,601. 

29. 

1- 

29. 

42m. 

Page  166. 

44. 

99,485. 

30. 

75$. 

30. 

182fff. 

17. 

41  pounds. 

45. 

2000. 

31. 

H 

31. 

157$f|. 

18. 

$  3.60. 

46. 

2800. 

32. 

6$. 

32. 

49f|-f. 

19. 

150  days  ;  3 

47. 

24,000. 

33. 

18$. 

33. 

25f£f. 

days. 

48. 

24,500. 

34. 

43f. 

34. 

40ft. 

20. 

$2.05. 

49. 

99,000. 

35. 

12f. 

35. 

30rm?. 

21. 

$39. 

50. 

96,000. 

36. 

llf. 

36. 

SS^s^ 

22. 

$225. 

51. 

81,081. 

37. 

24*. 

37. 

5<fV9oV 

23. 

$2.25. 

38. 

18*. 

38. 

12****. 

24. 

$  8.22. 

Page  161. 

39. 

27*. 

39. 

19*{|**. 

25. 

§40. 

1. 

129$. 

40. 

40f. 

40. 

14****. 

2. 

92*. 

41. 

23ii|.             page  169. 

3. 

79|. 

1. 

2857^T. 

42. 

VtfoV 

1. 

603,275. 

4. 

97f. 

2. 

3134$f 

43. 

3*ffft 

2. 

678,456. 

5. 

69$. 

3. 

1225$f 

44. 

17/j2/j. 

3. 

759,795. 

€. 

27*. 

4. 

1622|f 

45. 

3T^ilT- 

4. 

641,426. 

7. 

47*. 

5. 

990/T. 

46. 

4JT17T- 

5. 

$2714.42. 

ANSWERS. 


15 


6.  $8502.43. 

45.  963,976. 

86.  3134ff. 

125.  34. 

7.  $7269.80. 

46.  887,112. 

87.  31425V 

126.  125. 

8.  $9885.02. 

47.  629,405. 

88.  3009¥\. 

127.  5. 

9.  300,424. 

48.  890,765. 

89.  3034ff. 

128.  138. 

10.  913,092. 

49.  933,725. 

90.  3050f£. 

129.  150. 

11.  $220,119. 

50.  2123. 

91.  30717V 

130.  78. 

12.  $1912.09. 

51.  1203. 

92.  2016Ty?. 

13.  $359,809. 

52.  1303. 

93.  1234if£. 

Page  172. 

14.  414,867. 

53.  1203. 

94.  1132f|f. 

1.  760  yards. 

15.  $161,715. 

54.  1031. 

95.  1355¥522j. 

2.  240hf.pt. 

16.  173,929. 

55.  2402. 

96.  504. 

3.  $3.45. 

17.  $2952.51. 

56.  3002. 

97.  306. 

4.  $248. 

18.  399,952. 

57.  3030. 

98.  203. 

5.  $210. 

19.  $1624.43. 

58.  10,444. 

99.  105. 

6.  4|  pounds. 

59.  1060. 

100.  109f$ff 

7.  41$  bushels. 

Page  170. 

60.  1011. 

8.  $4.98. 

20.  868,980. 

61.  1012. 

Page  171. 

21.  895,048. 

62.  1013. 

101.  59$. 

Page  173. 

22.  954,048. 

63.  1011. 

102.  19$. 

9.  1$  minutes. 

23.  996,450. 

64.  1101. 

103.  9639. 

10.  1440  matches. 

24.  592,320. 

65.  1102.   » 

104.  12,141. 



25.  864,128. 

66.  220. 

105.  96. 

1.  5f. 

26.  970,485. 

67.  303. 

106.  96. 

2.  12$. 

27.  940,215. 

68.  150. 

107.  12$. 

3.  15f. 

28.  967,890. 

69.  606. 

108.  60|. 

4.  20. 

29.  954,087. 

70.  222. 

109.  61$. 

5.  27$. 

30.  906,205. 

71.  300|ff|. 

110.  300. 

6.  42f. 

31.  968,464. 

72.  306|^|. 

111.  300. 

7.  38$. 

32.  886,730. 

73.  219. 

112.  2. 

8.  56$. 

33.  864,565. 

74.  lOlffff. 

113.  8. 

9.  134$. 

34.  941,408. 

75.  154|§ff. 

114.  78. 

10.  134}. 

35.  948,708. 

76.  112fK-f. 

115.  162. 

11.  88$. 

36.  972,930. 

77.  112$fjff. 

116.  231. 

12.  50. 

37.  761,472. 

78.  861^V 

117.  36. 

13.  44$. 

38.  955,320. 

79.  833  J»¥. 

118.  648. 

14.  26f. 

39.  969,855. 

80.  903^. 

119.  46$. 

15.  36|. 

40.  976,372. 

81.  982$f. 

120.  70f. 

16.  9f. 

41.  926,328. 

82.  1313$|. 

121.  12,126. 

17.  46}. 

42.  925,245. 

83.  2196ff. 

122.  187,440. 

18.  147$. 

43.  856,674. 

84.  2218ff. 

123.  68. 

19.  37$. 

44.  977,724. 

85.  2279£|. 

124.  975. 

20.  73$. 

16 


ANSWERS. 


21.   7J. 

24.   969,600. 

3.   39ft. 

42.    70J. 

22.   4. 

25.   617,120. 

4.    70ft. 

43.   3S&. 

23.    86f. 

26.  434,420. 

5.   81. 

44.    7J. 

24.    29£. 

27.   47,196. 

6.   9ft. 

45.   9|. 

25.   34f 

28.   47,272. 

7.   21ft. 

46.   8A. 

26.   59$. 

29.   47,082. 

8.    7ft. 

47.    78J. 

27.   61|. 

30.   137,598. 

9.    92ft. 

48.   55TV 

28.   m. 

31.    59,660. 

10.   57f. 

49.    47A- 

29.   17f 

32.   59,508. 

50.   39ft. 

30.   7i 

33.    137,427. 

Page  177. 

31.    8|. 

34.    78,150. 

11-   ft* 

Page  179. 

32.    18f 

35.    209,664. 

12.   2ft. 

1.   3210. 

33.    49|. 

36.    844,662. 

13.    4ft. 

2.   4321. 

34.    25£. 

37.   979,016. 

14.    4ft. 

3.    765. 

35.    74£. 

38.   998,016. 

15.    18ft. 

4.   3450T$¥. 

36.    23J. 

39.    17,329f 

16.   33ft. 

5.  5403. 

40.   58,8671. 

17.   53ft. 

6.   4506Tf*. 

Page  174. 

41.    760,249. 

18.    76. 

7.   6063. 

1.   46,512. 

42.   369,123. 

19.    105. 

8.   7006. 

2.    144,536. 

43.   17,367f 

20.   48ft. 

9.   6003. 

3.   253,840. 

44.   30,250. 

21.    9ft. 

10.  6005. 

4.    132,435. 

45.   850,950. 

22.   61&. 

11.   7001. 

5.   306,130. 

46.  95,482$. 

23.   63&. 

12.   5203. 

6.   354,488. 

47.   938,475. 

24.    7ft. 

13.   6715T6&. 

7.    87,084. 

48.    935,712. 

25.    15&. 

14.   5701. 

8.    199,014. 

49.   781,1371. 

26.    32^. 

15.    1020TVT. 

9.    784,770. 

50.  954,320. 

27.   29ft. 

16.   2034. 

10.   934,164. 

28.   29ft. 

17.   3240. 

11.   784,770. 

29.   82. 

18.   4003. 

12.   934,164. 

30.    82f 

19.   5041T^. 

13.   30,504. 

31.   21J. 

20.   4774TVf. 

14.   54,756. 

32.   31f 

21.    1789^. 

15.   37,260. 

33.    41  f. 

22.   1509=£fr. 

16.   138,624. 

34.   51f. 

23.   1155ft£ 

17.   616,302. 

35.    61f. 

24.    263  H&f 

18.   104,148. 

36.    61f 

25.    2347iff- 

19.   805,460. 

37.    71f 

26.   298  If  ft. 

20.   93,912. 

38.    84*. 

27.    1435^. 

21.    151,782. 

Page  176. 

39.   79TV 

28.   4f>9?i4. 

22.    548,730. 

1.    20^. 

40.    65^. 

29.    1545fff. 

23.    846,300. 

2.   25ft. 

41.   59f 

30.    720^°T. 

ANSWERS. 


17 


31. 

2117^ft. 

3. 

45**- 

44. 

31f 

Page  189. 

32. 

1707ff*. 

4. 

66ft. 

45. 

13*- 

4. 

$4. 

33. 

1607*H. 

5. 

79ft- 

46. 

32|. 

5. 

29  tons. 

34. 

1615***. 

6. 

75ft. 

47. 

13*. 

6. 

195  days. 

35. 

1191ft*. 

7. 

17ft. 

48. 

14*. 

7. 

13  cents. 

36. 

1053***. 

8. 

16ft. 

49. 

.41*. 

8. 

416  yards. 

37. 

1008***, 

9. 

49ft. 

50. 

20ft. 

9. 

$405. 

38. 

879f§£. 

10. 

38ft. 

10. 

537  pounds. 

39. 

990Hf 

11. 

18ft. 

11. 

35  plants. 

40. 

6oo?yT. 

12. 

42ft. 

Page  186. 

12. 

341  passengers. 

41. 

608Tyft. 

13. 

65ft. 

1. 

117  ounces. 

13. 

$  3000. 

42. 

461^5. 

14. 

29H- 

2. 

4  Ib.  5  oz. 

14. 

Lost  $  20. 

43. 

307fl&. 

15. 

75H- 

3. 

20  gal.  2  qt. 

15. 

1799. 

44. 

185|f£f. 

16. 

116*. 

4. 

59  quarts. 

16. 

8  years. 

45. 

153|£*|. 

17. 

50ft. 

5. 

23  qt.  1  pt. 

46. 

25**ff 

18. 

65  J. 

6. 

57  pints. 

Page  190. 

47. 

7()2|9|. 

19. 

92*. 

7. 

75  pecks. 

17. 

$3. 

48. 

30ff|f. 

20. 

97ft. 

8. 

143  quarts. 

18. 

$420. 

49. 

32***f 

21. 

28*. 

9. 

12  pk.  1  qt. 

19. 

$225. 

50. 

283||**. 

22. 

59. 

10. 

21  bu.  3  pk. 

20. 

46  boys. 

51. 

2302%°^T. 

23. 

12*. 

11. 

1568  quarts. 

21. 

$2. 

52. 

251***f. 

24. 

99ft. 

12. 

180  inches. 

22. 

$216. 

25. 

96f. 

13. 

44  feet. 

23. 

10  cents. 

Page  181. 

26. 

251 

14. 

159  inches. 

24. 

9  months. 

1. 

240  bushels. 

27. 

71*. 

15. 

9  ft.  11  in. 

25. 

200  eggs. 

2. 

58  cents. 

28. 

56*. 

16. 

23  yd.  1  ft. 



3. 

5  cows. 

29. 

25ft. 

17. 

44  pounds. 

1. 

835,539. 

4. 

90  cents. 

30. 

3ft- 

18. 

88  gallons. 

2. 

759,645. 

5. 

$5500. 

31. 

14*. 

19. 

65  quarts. 

3. 

888,732. 

6. 

226£  acres. 

32. 

59*. 

20. 

151  bushels. 

4. 

869,649. 

7. 

$5.88. 

33. 

38|. 

21. 

43  pecks. 

5. 

805,050. 

8. 

$90. 

34. 

31*. 

22. 

85  ft.  3  in. 

6. 

746,108. 

9. 

402. 

35. 

18i. 

23. 

60  bu.  3  pk. 

7. 

902,000. 

10. 

4  cows. 

36. 

31*. 

24. 

13  ft.  1  in. 

8. 

963,214. 

11. 

16  days. 

37. 

30*. 

25. 

67  gal.  2  qt. 

9. 

855,922. 

12. 

3  yards. 

38. 

22f. 

10. 

957,032. 

13. 

80  quarts. 

39. 

41*. 

11. 

704,175. 

40. 

37*. 

Page  188. 

12. 

593,164. 

Page  185. 

41. 

13*. 

1. 

80. 

13. 

986,592. 

1. 

18ft- 

42. 

3*. 

2. 

$1200;  $240. 

14. 

962,304. 

2. 

24ft. 

43. 

27*. 

3. 

$1. 

15. 

943,114. 

18 


ANSWERS. 


16.  831,875. 

55.  1046ff. 

96.  144||f. 

17.  833,316. 

56.  1033$f. 

97.  821|f$. 

18.  505,134. 

57.  841f|. 

98.  91TVFV 

58.  6G9|f. 

99.  241ff||. 

Page  191. 

59.  215ff. 

100.  237|1§|. 

19.  190|. 

60.  223AV 

101.  63^V 

20.  500. 

61.  260|||. 

102.  181|||f. 

21.  420. 

62.  89f|f. 

22.  1845. 

63.  83$$|. 

23.  987. 

64.  40ff$. 

Page  192. 

24.  1071. 

65.  78^7- 

103.  97. 

25.  1612. 

66.  81^^. 

104.  48. 

26.  1G45. 

67.  32f|f. 

105.  32. 

27.  2583. 

68.  99$$$. 

106.  24. 

28.  3885. 

69.  43$|f. 

107.  18. 

29.  4100. 

*y  f\  o£  A  4  9 
iu.  ^^^rlJoT' 

108.  16. 

30.  780,096. 

n9^1  511 
•   .-OOy  tJirT. 

109.  14. 

31.  991,782. 

72.  281|ff|. 

110.  12. 

32.  943,260. 

70   17Q2807 
/O.   1  /  <?J5§7. 

111.  11. 

33.  984,328. 

74.  166flft. 

112.  44. 

34.  892,320. 

75.  51325%V 

113.  33. 

35.  952,408. 

76.  107£f££. 

114.  22. 

36.  933,450. 

77.  207$|f. 

37.  875,706. 

78.  SOffff. 

38.  952,714. 

*7Q   QA6  342 
/  «7.  ^^§^^X» 

Page  193. 

39.  970,169. 

80.  579$|. 

1.  1*. 

40.  954,530. 

81.  2332ff 

2.  4TV 

41.  3519. 

82.  767$f. 

3.  7TV 

42.  3616. 

83.  628ff. 

d   Ifi1  ! 
Tt.  J.\J  j  *. 

43.  6132. 

84.  1398|f. 

5.  37if. 

44.  4557. 

85.  10217f. 

6.  ISA. 

45.  9568. 

86.  1051ff. 

7.  11  A- 

46.  10,791. 

87.  974f£. 

8.  31  A- 

47.  17,572. 

88.  lOS^ff 

9-   8yj- 

48.  39,333. 

89.  27822jV 

10.  71A- 

49.  76,775. 

90.  841H. 

11.  lsf$- 

50.  97,460. 

91.  184TVV 

12.  26if. 

51.  69,000. 

92.  905^. 

13  53$. 

52.  2218|f. 

93.  554fff. 

14.  991. 

53.  786i|. 

94.  951  f|£. 

15.  911. 

54.  1618J$. 

95.  285|||. 

16.  4  7  A- 

17.  77^. 

18.  86H- 

19.  54. 

20.  63$. 

21.  88^. 

22.  93|. 

23.  94f. 

24.  93f 

25.  81$J. 

26.  SI 

27.  76H- 

28.  88$. 
29. 

30. 

31.  71$. 

32.  3f. 

33.  24f. 

34.  18f. 

35.  21|. 
36. 

37. 
38. 
39. 
40. 


ANSWERS. 


19 


6.  594,672. 

36.  577,771. 

Page  197. 

7.  531,696. 

37.  194,142. 

69.  59$ff. 

8.  178,654. 

38.  806,922. 

70.  18$f|. 

9.  194,508. 

39.  834,725. 

71.  Slfff 

10.  177,045. 

40.  696,822. 

72.  123fff. 

11.  329,141. 

41.  92,501$. 

73.  21$H. 

12.  537,966. 

42.  205,979. 

74.  33^f. 

43.  301,058. 

75.  3223iff. 

44.  293,336|. 

76.  370j%V 

Page  196. 

45.  397,087. 

77.  410iff 

13.  503,036. 

46.  564,389f 

78.  908fff. 

14.  354,585. 

47.  378,670$. 

79.  930$|f. 

15.  348,087. 

48.  489,303$. 

80.  460|f|. 

16.  781,529. 

49.  571,693$. 

81.  417fff. 

17.  75,854. 

50.  352,315$. 

82.  263f£f. 

18.  63,616f. 

51.  3646$f 

83.  255  J  fff. 

19.  56,818$. 

52.  2376$f 

84.  197iflf. 

20.  80,647$. 

53.  1002|f 

85.  194ff$f. 

21.  77,371f. 

54.  1578ff 

86.  116ff|f. 

22.  114,608f. 

55.  326£f 

87.  54fff|. 

23.  220,676. 

56.  711W- 

88.  68^1- 

24.  313,985$. 

57.  361$f 

89.  29fff£. 

25.  434,661f. 

58.  441  /T. 

90.  76|fff. 

26.  447,673. 

59.  3297|$. 

91.  135$$|f. 

27.  488,748. 

60.  977$|. 

92.  228$$ff. 

28.  551,5361. 

61.  2159$|. 

93.  29||f|. 

29.  486,029|. 

62.  311ff. 

94.  119ffff 

Page  195. 

30.  881,001$. 

63.  1157&- 

95.  47f§§|. 

1.  260,512. 

31.  101,376. 

64.  10737V 

96.  97ff|f. 

2.  110,124. 

32.  162,582. 

65.  1048$$. 

97.  50f§ff. 

3.  387,024. 

33.  145,116. 

66.  314ff. 

98.  63?W?> 

4.  237,394. 

34.  130,245. 

67.  63|f|. 

99.  72%Y7. 

5.  505,580. 

35.  84,632. 

68.  172//J. 

100.  5$ff$§. 

Elementary   Mathematics 


AtWQOd's  Complete  Graded  Arithmetic.     Presents  a  carefully  graded  course,  to 

begin  with  the  fourth  year  and  continue  through  the  eighth  year.    Part  I,  30  cts.;  Part 

II,  65  cts. 
Badlam's  Aids  tO  Number.      Teacher's  edition  —  First  series,  Nos.  i  to  10,  40  cts.; 

Second   series,  Nos.  10  to  20,  40  cts.    Pupil's  edition — First  series,  25  cts.;  Second 

series,  25  cts. 

Branson's  Methods  in  Teaching  Arithmetic.     15  cts. 

Hanus's  Geometry  in  the  Grammar  Schools.    An  essay,  with  outline  of  work  for 

the  last  three  years  of  the  grammar  school.     25  cts. 

Howland's  Drill  Cards.  For  middle  grades  in  arithmetic.  Each,  3  cts.;  per  hun- 
dred, $2.40. 

Hunt's  Geometry  for  Grammar  SchOOls.  The  definitions  and  elementary  con- 
cepts are  to  be  taught  concretely,  by  much  measuring,  and  by  the  making  of  models 
and  diagrams  by  the  pupils.  30  cts. 

PierCC'S  Review  Number  Cards.       Two  cards,   for  second  and   third  year  pupils. 

Each,  3  cts.;  per  hundred,  $2.40. 
Safford's  Mathematical  Teaching.      A  monograph,  with  applications.    25  cts. 

Sloane's  Practical  Lessons  in  Fractions.     25  cts.    Set  of  six  fraction  cards,  fot 

pupils  to  cut.     10  cts. 

Sutton  and  Kimbrough's  Pupils'  Series  of  Arithmetics.      Lower  Book,  for 

primary  and  intermediate  grades,  35  cts.     Higher  Book,  65  cts. 

The  New  Arithmetic.  By  300  teachers.  Little  theory  and  much  practice.  An  excel- 
lent review  book.  65  cts. 

Walsh's  Arithmetics.  On  the,  "spiral  advancement"  plan,  and  perfectly  graded. 
Special  features  of  this  series  are  its  division  into  half-yearly  chapters  instead  of  the 
arrangement  by  topics;  the  great  number  and  variety  of  the  problems ;  the  use  of  the 
equation  in  solution  of  arithmetical  problems;  and  the  introduction  of  the  elements  of 
algebra  and  geometry.  Its  use  shortens  and  enriches  the  course  in  common  school 
mathematics.  In  two  series:  — 

Three  Book  Series — Elementary,  30  cts.;  Intermediate,  35  cts.;  Higher, 65  cts. 
Two  Book  Series  —  Primary,  30  cts.;  Grammar  school   65  cts. 

Walsh's  Algebra  and  Geometry  for  Grammar  Grades.     Three  chapters  from 

Walsh's  Arithmetic  printed  separately.     15  cts. 

White's  TWO  Years  With  Numbers.      For  second  and  third  year  classes.     35  cts. 
White's  Junior  Arithmetic.      For  fourth  and  fifth  years.     45  cts. 

White's  Senior  Arithmetic.     65  cts. 

For  advanced  -works  see  our  list  of  books  in  Mathematics. 

D.C.  HEATH  &  CO.,  Publishers,  Boston, New  York, Chicago 


Elementary  Science. 


Austin's  Observation  Blanks  in  Mineralogy.     Detailed  studies  of  3S  minerals. 

Boards.     88  pages.     30  cts. 

Bailey's  Grammar  School  Physics.  A  series  of  inductive  lessons  in  the  elements 
of  the  science.  Illustrated.  60  cts. 

Ballard'S  The  World  Of  Matter.  A  guide  to  the  study  of  chemistry  and  mineralogy; 
adapted  to  the  general  reader,  for  use  as  a  text-book  or  as  a  guide  to  the  teacher  in  giving 
object-lessons.  264  pages.  Illustrated.  $1.00. 

Clark's  Practical  Methods  in  MicrOSCOpy.  Gives  in  detail  descriptions  of  methods 
that  will  lead  the  careful  worker  to  successful  results.  233  pages.  Illustrated.  $1.60. 

Clarke's  Astronomical  Lantern.  Intended  to  familiarize  students  with  the  constella- 
tions by  comparing  them  with  fac-similes  on  the  lantern  face.  With  seventeen  slides, 
giving  twenty-two  constellations.  $4  50. 

Clarke's  HOW  tO  find  the  Stars.  Accompanies  the  above  and  helps  to  an  acquaintance 
with  the  constellations.  47  pages.  Paper.  15  cts. 

Guides  for  Science  Teaching.  Teachtrs'  aids  in  the  instruction  of  Natural  History 
classes  in  the  lower  p rades. 

I.     Hyatt's  About  Pebbles.     a6  pages.     Paper.     10  cts. 
II.     Goodale's  A  Few  Common  Plants.     61  pages.     Paper.     20  cts. 
III.     Hyatt's  Commercial  and  other  Sponges.    Illustrated.    43  pages.  Paper.   20  cts. 


IV.    Agassir's  First  Lessons  in  Natural  History.     Illustrated.     64  pages.     Paper. 

25  cts. 

V.     Hyatt's  Corals  and  Echinoderms.     Illustrated.     32  pages.    Paper.     30  cts. 
VI.     Hyatt's  Mollusca.     Illustrated,     f  5  pages.     Paper.     30  cts. 
VII.     Hyat's  Worms  and  Crustacea.     Illustrated.     68  pages.     Paper.     30  cts. 
VIII.     Hyatt's  Insecta.     Illustrated.     324  pages.     Cloth.     $1.25. 
XII.     Crosby's  Common  Minerals  aad  Rocks.     Illustrated.     2*0  pages.     Paper,  40 
cts.     Cloth,  60  cts. 

XIII.  Richard's  First  Lessons  in  Minerals.     50  pages.     Paper.     10  cts. 

XIV.  Bowditch's  Physiology.     58  pages.     Paper.     20  cts. 

XV.     Clapp's  36  Observation  Lessons  in  Minerals.     80  pages.     Paper.     30  cts. 
XVI.     Phe MY 's  Lessons  in  Chemistry.    20  cts. 
Pupils'  Note- Book  to  accompany  No.  15.     10  cts. 

Rice's  Science  Teaching  in  the  School.  With  a  course  of  instruction  in  science 
for  the  lower  grades.  46  p*g  s.  Paper.  25  cts. 

Ricks's  Natural  History  Object  LeSSOnS.  Supplies  information  on  plants  and 
their  products,  on  animals  and  their  uses,  and  gives  specimen  lessons.  Fully  illustrated 
332  pages.  $1.50. 

Ricks's  Object  Lessons  and  How  to  Give  them. 

Volume  I.     Gives  lessons  for  primary  grades.     200  pages.     90  cts. 

Volume  II.  Gives  lessons  for  grammar  and  intermediate  grades.  212  pages.     90  cts. 

Staler  S  First  Book  in  Geology.  For  high  school,  or  highest  class  in  grammar  school 
272  pages.  Illustrated.  $1.00. 

Shaler'S  Teacher's  Methods  in  Geology.  An  aid  to  the  teacher  of  Geology. 
74  pages.  Paper.  25  cts. 

Smith's  Studies  in  Nature.  A  combination  of  natural  history  lessons  and  language 
work.  48  pages.  Paper.  1 5  cts. 

See  also  our  list  of  books  in  Science. 

D.  C.  HEATH  &CO.,Publishers,Boston,  New  York,  Chicago 


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